Solve for: integral of (7x-15)/(x^3+2x^2+5x) x

Expression: $\int{ \frac{ 7x-15 }{ {x}^{3}+2{x}^{2}+5x } } \mathrm{d} x$

Rewrite the fraction using partial-fraction decomposition

$\int{ -\frac{ 3 }{ x }+\frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$

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

$-\int{ \frac{ 3 }{ x } } \mathrm{d} x+\int{ \frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$

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

$-3\ln\left({|x|}\right)+\int{ \frac{ 3x+13 }{ {x}^{2}+2x+5 } } \mathrm{d} x$

Evaluate the indefinite integral

$-3\ln\left({|x|}\right)+\frac{ 3 }{ 2 } \times \ln\left({|{x}^{2}+2x+5|}\right)+5\arctan\left({\frac{ x+1 }{ 2 }}\right)$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }-3\ln\left({|x|}\right)+\frac{ 3 }{ 2 } \times \ln\left({|{x}^{2}+2x+5|}\right)+5\arctan\left({\frac{ x+1 }{ 2 }}\right)+C,& C \in ℝ\end{array}$

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