Optimal. Leaf size=57 \[ \frac{d}{b^2 (b+c x)}-\frac{d \log (b+c x)}{b^3}+\frac{d \log (x)}{b^3}+\frac{c d-b e}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.0437102, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{d}{b^2 (b+c x)}-\frac{d \log (b+c x)}{b^3}+\frac{d \log (x)}{b^3}+\frac{c d-b e}{2 b c (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d}{b^3 x}+\frac{-c d+b e}{b (b+c x)^3}-\frac{c d}{b^2 (b+c x)^2}-\frac{c d}{b^3 (b+c x)}\right ) \, dx\\ &=\frac{c d-b e}{2 b c (b+c x)^2}+\frac{d}{b^2 (b+c x)}+\frac{d \log (x)}{b^3}-\frac{d \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0416078, size = 53, normalized size = 0.93 \[ \frac{\frac{b \left (b^2 (-e)+3 b c d+2 c^2 d x\right )}{c (b+c x)^2}-2 d \log (b+c x)+2 d \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 59, normalized size = 1. \begin{align*}{\frac{d\ln \left ( x \right ) }{{b}^{3}}}-{\frac{e}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{d}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{d\ln \left ( cx+b \right ) }{{b}^{3}}}+{\frac{d}{{b}^{2} \left ( cx+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10209, size = 92, normalized size = 1.61 \begin{align*} \frac{2 \, c^{2} d x + 3 \, b c d - b^{2} e}{2 \,{\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )}} - \frac{d \log \left (c x + b\right )}{b^{3}} + \frac{d \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76518, size = 236, normalized size = 4.14 \begin{align*} \frac{2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e - 2 \,{\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (c x + b\right ) + 2 \,{\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{3} x^{2} + 2 \, b^{4} c^{2} x + b^{5} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.89887, size = 63, normalized size = 1.11 \begin{align*} \frac{- b^{2} e + 3 b c d + 2 c^{2} d x}{2 b^{4} c + 4 b^{3} c^{2} x + 2 b^{2} c^{3} x^{2}} + \frac{d \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12754, size = 81, normalized size = 1.42 \begin{align*} -\frac{d \log \left ({\left | c x + b \right |}\right )}{b^{3}} + \frac{d \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e}{2 \,{\left (c x + b\right )}^{2} b^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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