Optimal. Leaf size=139 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
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Rubi [A] time = 0.122614, antiderivative size = 139, 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 = {698} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac{c^2}{e^4}+\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^3}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^2}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{c^2 x}{e^4}-\frac{\left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 (d+e x)^2}-\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^5 (d+e x)}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0832742, size = 176, normalized size = 1.27 \[ \frac{-e^2 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c e \left (b d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 a e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 279, normalized size = 2. \begin{align*}{\frac{{c}^{2}x}{{e}^{4}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,abd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{ab}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+2\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{4}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993387, size = 262, normalized size = 1.88 \begin{align*} -\frac{13 \, c^{2} d^{4} - 11 \, b c d^{3} e + a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{2 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94339, size = 578, normalized size = 4.16 \begin{align*} \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - a b d e^{3} - a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - b c d^{3} e +{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \,{\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.45517, size = 218, normalized size = 1.57 \begin{align*} \frac{c^{2} x}{e^{4}} + \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{2} e^{4} + a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 11 b c d^{3} e + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 3 b^{2} e^{4} - 18 b c d e^{3} + 18 c^{2} d^{2} e^{2}\right ) + x \left (3 a b e^{4} + 6 a c d e^{3} + 3 b^{2} d e^{3} - 27 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10542, size = 230, normalized size = 1.65 \begin{align*} c^{2} x e^{\left (-4\right )} - 2 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + a b d e^{3} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + a^{2} e^{4} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} + a b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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