Optimal. Leaf size=137 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{c^2}{3 e^5 (d+e x)^3} \]
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Rubi [A] time = 0.0880271, 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} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{c^2}{3 e^5 (d+e x)^3} \]
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)^8} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^8}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^7}+\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^6}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^5}+\frac{c^2}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac{d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac{d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac{6 c^2 d^2-6 b c d e+b^2 e^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0477294, size = 117, normalized size = 0.85 \[ -\frac{2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 143, normalized size = 1. \begin{align*} -{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{c \left ( be-2\,cd \right ) }{2\,{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 = 1.16599, size = 279, normalized size = 2.04 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56862, size = 435, normalized size = 3.18 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.3192, size = 218, normalized size = 1.59 \begin{align*} - \frac{2 b^{2} d^{2} e^{2} + 3 b c d^{3} e + 2 c^{2} d^{4} + 70 c^{2} e^{4} x^{4} + x^{3} \left (105 b c e^{4} + 70 c^{2} d e^{3}\right ) + x^{2} \left (42 b^{2} e^{4} + 63 b c d e^{3} + 42 c^{2} d^{2} e^{2}\right ) + x \left (14 b^{2} d e^{3} + 21 b c d^{2} e^{2} + 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25388, size = 180, normalized size = 1.31 \begin{align*} -\frac{{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 2 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{210 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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