Optimal. Leaf size=131 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{c^2 \log (d+e x)}{e^5} \]
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Rubi [A] time = 0.091824, antiderivative size = 131, 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 = {698} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{2 e^5 (d+e x)^2}-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{c^2 \log (d+e x)}{e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^5}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^4}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^3}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^2}+\frac{c^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{d^2 (c d-b e)^2}{4 e^5 (d+e x)^4}+\frac{2 d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{2 e^5 (d+e x)^2}+\frac{2 c (2 c d-b e)}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0447406, size = 126, normalized size = 0.96 \[ \frac{-b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )-6 b c e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 197, normalized size = 1.5 \begin{align*} -{\frac{{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}+{\frac{2\,{b}^{2}d}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-2\,{\frac{bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-2\,{\frac{bc}{{e}^{4} \left ( ex+d \right ) }}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{bc{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17247, size = 239, normalized size = 1.82 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - b^{2} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \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 [A] time = 1.57932, size = 456, normalized size = 3.48 \begin{align*} \frac{25 \, c^{2} d^{4} - 6 \, b c d^{3} e - b^{2} d^{2} e^{2} + 24 \,{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \,{\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - b^{2} e^{4}\right )} x^{2} + 4 \,{\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - b^{2} d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.06712, size = 180, normalized size = 1.37 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} - \frac{b^{2} d^{2} e^{2} + 6 b c d^{3} e - 25 c^{2} d^{4} + x^{3} \left (24 b c e^{4} - 48 c^{2} d e^{3}\right ) + x^{2} \left (6 b^{2} e^{4} + 36 b c d e^{3} - 108 c^{2} d^{2} e^{2}\right ) + x \left (4 b^{2} d e^{3} + 24 b c d^{2} e^{2} - 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34748, size = 289, normalized size = 2.21 \begin{align*} -c^{2} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{24 \, b c e^{16}}{x e + d} + \frac{36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac{24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac{6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac{6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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