Optimal. Leaf size=96 \[ -\frac{3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac{d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c d-b e}{e^4 (d+e x)}+\frac{c \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.0778137, antiderivative size = 96, 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} \[ -\frac{3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac{d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c d-b e}{e^4 (d+e x)}+\frac{c \log (d+e x)}{e^4} \]
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)^4} \, dx &=\int \left (-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^4}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^3}+\frac{-3 c d+b e}{e^3 (d+e x)^2}+\frac{c}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^3}-\frac{3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac{3 c d-b e}{e^4 (d+e x)}+\frac{c \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0368533, size = 86, normalized size = 0.9 \[ \frac{-e \left (a e (d+3 e x)+2 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c d \left (11 d^2+27 d e x+18 e^2 x^2\right )+6 c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 128, normalized size = 1.3 \begin{align*}{\frac{c\ln \left ( ex+d \right ) }{{e}^{4}}}+{\frac{ad}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{b{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{d}^{3}c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{a}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{bd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,c{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{b}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{cd}{{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 = 1.02965, size = 153, normalized size = 1.59 \begin{align*} \frac{11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{c \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 = 1.28395, size = 296, normalized size = 3.08 \begin{align*} \frac{11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \,{\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x + 6 \,{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.59645, size = 114, normalized size = 1.19 \begin{align*} \frac{c \log{\left (d + e x \right )}}{e^{4}} - \frac{a d e^{2} + 2 b d^{2} e - 11 c d^{3} + x^{2} \left (6 b e^{3} - 18 c d e^{2}\right ) + x \left (3 a e^{3} + 6 b d e^{2} - 27 c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08241, size = 119, normalized size = 1.24 \begin{align*} c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (6 \,{\left (3 \, c d e - b e^{2}\right )} x^{2} + 3 \,{\left (9 \, c d^{2} - 2 \, b d e - a e^{2}\right )} x +{\left (11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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