Calculate: integral of (x^3)/(x^2+25) x

Expression: $\int{ \frac{ {x}^{3} }{ {x}^{2}+25 } } \mathrm{d} x$

Use the substitution $t={x}^{2}+25$ to transform the integral

$\int{ \frac{ t-25 }{ 2t } } \mathrm{d} t$

Use the property of integral $\begin{array} { l }\int{ a \times f\left( x \right) } \mathrm{d} x=a \times \int{ f\left( x \right) } \mathrm{d} x,& a \in ℝ\end{array}$

$\frac{ 1 }{ 2 } \times \int{ \frac{ t-25 }{ t } } \mathrm{d} t$

Separate the fraction into $2$ fractions

$\frac{ 1 }{ 2 } \times \int{ \frac{ t }{ t }-\frac{ 25 }{ t } } \mathrm{d} t$

Any expression divided by itself equals $1$

$\frac{ 1 }{ 2 } \times \int{ 1-\frac{ 25 }{ t } } \mathrm{d} t$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\frac{ 1 }{ 2 } \times \left( \int{ 1 } \mathrm{d} t-\int{ \frac{ 25 }{ t } } \mathrm{d} t \right)$

Use $\int{ 1 } \mathrm{d} x=x$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( t-\int{ \frac{ 25 }{ t } } \mathrm{d} t \right)$

Use $\int{ \frac{ a }{ x } } \mathrm{d} x=a \times \ln\left({|x|}\right)$ to evaluate the integral

$\frac{ 1 }{ 2 } \times \left( t-25\ln\left({|t|}\right) \right)$

Substitute back $t={x}^{2}+25$

$\frac{ 1 }{ 2 } \times \left( {x}^{2}+25-25\ln\left({|{x}^{2}+25|}\right) \right)$

Simplify the expression

$\frac{ 1 }{ 2 }{x}^{2}+\frac{ 25 }{ 2 }-\frac{ 25 }{ 2 } \times \ln\left({{x}^{2}+25}\right)$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\frac{ 1 }{ 2 }{x}^{2}-\frac{ 25 }{ 2 } \times \ln\left({{x}^{2}+25}\right)+C,& C \in ℝ\end{array}$