Optimal. Leaf size=89 \[ \frac{d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac{(3 c d-b e) \log (d+e x)}{e^4}+\frac{c x}{e^3} \]
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Rubi [A] time = 0.0834127, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ \frac{d \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac{(3 c d-b e) \log (d+e x)}{e^4}+\frac{c x}{e^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{x \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac{c}{e^3}-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^3}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^2}+\frac{-3 c d+b e}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{c x}{e^3}+\frac{d \left (c d^2-b d e+a e^2\right )}{2 e^4 (d+e x)^2}-\frac{3 c d^2-e (2 b d-a e)}{e^4 (d+e x)}-\frac{(3 c d-b e) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.060328, size = 80, normalized size = 0.9 \[ \frac{-\frac{2 \left (e (a e-2 b d)+3 c d^2\right )}{d+e x}+\frac{d e (a e-b d)+c d^3}{(d+e x)^2}+2 (b e-3 c d) \log (d+e x)+2 c e x}{2 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 121, normalized size = 1.4 \begin{align*}{\frac{cx}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) b}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) cd}{{e}^{4}}}+{\frac{ad}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{b{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{d}^{3}c}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{a}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{bd}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{c{d}^{2}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0626, size = 130, normalized size = 1.46 \begin{align*} -\frac{5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \,{\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{c x}{e^{3}} - \frac{{\left (3 \, c d - b e\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34546, size = 308, normalized size = 3.46 \begin{align*} \frac{2 \, c e^{3} x^{3} + 4 \, c d e^{2} x^{2} - 5 \, c d^{3} + 3 \, b d^{2} e - a d e^{2} - 2 \,{\left (2 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x - 2 \,{\left (3 \, c d^{3} - b d^{2} e +{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 2 \,{\left (3 \, c d^{2} e - b d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0137, size = 97, normalized size = 1.09 \begin{align*} \frac{c x}{e^{3}} - \frac{a d e^{2} - 3 b d^{2} e + 5 c d^{3} + x \left (2 a e^{3} - 4 b d e^{2} + 6 c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac{\left (b e - 3 c d\right ) \log{\left (d + e x \right )}}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08069, size = 111, normalized size = 1.25 \begin{align*} c x e^{\left (-3\right )} -{\left (3 \, c d - b e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, c d^{3} - 3 \, b d^{2} e + a d e^{2} + 2 \,{\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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