Optimal. Leaf size=119 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c (2 c d-b e)}{e^4 (d+e x)}+\frac{2 c^2 \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.0947904, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac{3 c (2 c d-b e)}{e^4 (d+e x)}+\frac{2 c^2 \log (d+e x)}{e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^4} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^4}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)^3}-\frac{3 c (2 c d-b e)}{e^3 (d+e x)^2}+\frac{2 c^2}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^3}-\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{2 e^4 (d+e x)^2}+\frac{3 c (2 c d-b e)}{e^4 (d+e x)}+\frac{2 c^2 \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0491838, size = 111, normalized size = 0.93 \[ \frac{-2 c e \left (a e (d+3 e x)+3 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )-b e^2 (2 a e+b (d+3 e x))+2 c^2 d \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 c^2 (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 188, normalized size = 1.6 \begin{align*} -{\frac{ac}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{bcd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{bc}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{{c}^{2}d}{{e}^{4} \left ( ex+d \right ) }}-{\frac{ab}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,acd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{2}d}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{b{d}^{2}c}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{c}^{2}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995319, size = 197, normalized size = 1.66 \begin{align*} \frac{22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} -{\left (b^{2} + 2 \, a c\right )} 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{2 \, c^{2} \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 = 1.70907, size = 375, normalized size = 3.15 \begin{align*} \frac{22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} -{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} -{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x + 12 \,{\left (c^{2} e^{3} x^{3} + 3 \, c^{2} d e^{2} x^{2} + 3 \, c^{2} d^{2} e x + c^{2} 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 = 3.79181, size = 158, normalized size = 1.33 \begin{align*} \frac{2 c^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a b e^{3} + 2 a c d e^{2} + b^{2} d e^{2} + 6 b c d^{2} e - 22 c^{2} d^{3} + x^{2} \left (18 b c e^{3} - 36 c^{2} d e^{2}\right ) + x \left (6 a c e^{3} + 3 b^{2} e^{3} + 18 b c d e^{2} - 54 c^{2} 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 = 1.17683, size = 166, normalized size = 1.39 \begin{align*} 2 \, c^{2} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (18 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + 3 \,{\left (18 \, c^{2} d^{2} - 6 \, b c d e - b^{2} e^{2} - 2 \, a c e^{2}\right )} x +{\left (22 \, c^{2} d^{3} - 6 \, b c d^{2} e - b^{2} d e^{2} - 2 \, a c d e^{2} - 2 \, a b e^{3}\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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