Optimal. Leaf size=137 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2} \]
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Rubi [A] time = 0.0921652, antiderivative size = 137, 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} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2} \]
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)^7} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^7}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^6}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^5}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^4}+\frac{c^2}{e^4 (d+e x)^3}\right ) \, dx\\ &=-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{4 e^5 (d+e x)^4}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{c^2}{2 e^5 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.040776, size = 116, normalized size = 0.85 \[ -\frac{b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 143, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{2\,d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{2\,c \left ( be-2\,cd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{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 = 1.15776, size = 258, normalized size = 1.88 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63062, size = 396, normalized size = 2.89 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.8266, size = 204, normalized size = 1.49 \begin{align*} - \frac{b^{2} d^{2} e^{2} + 2 b c d^{3} e + 2 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (40 b c e^{4} + 40 c^{2} d e^{3}\right ) + x^{2} \left (15 b^{2} e^{4} + 30 b c d e^{3} + 30 c^{2} d^{2} e^{2}\right ) + x \left (6 b^{2} d e^{3} + 12 b c d^{2} e^{2} + 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28522, size = 178, normalized size = 1.3 \begin{align*} -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 6 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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