Optimal. Leaf size=117 \[ -\frac{c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac{4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac{c^2}{2 e^5 (d+e x)^2}+\frac{4 c^2 d}{3 e^5 (d+e x)^3} \]
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Rubi [A] time = 0.0690073, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ -\frac{c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac{4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac{c^2}{2 e^5 (d+e x)^2}+\frac{4 c^2 d}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^2}{(d+e x)^7} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^7}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^6}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^5}-\frac{4 c^2 d}{e^4 (d+e x)^4}+\frac{c^2}{e^4 (d+e x)^3}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac{4 c d \left (c d^2+a e^2\right )}{5 e^5 (d+e x)^5}-\frac{c \left (3 c d^2+a e^2\right )}{2 e^5 (d+e x)^4}+\frac{4 c^2 d}{3 e^5 (d+e x)^3}-\frac{c^2}{2 e^5 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0302626, size = 89, normalized size = 0.76 \[ -\frac{5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 120, normalized size = 1. \begin{align*} -{\frac{c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{4\,{c}^{2}d}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{4\,cd \left ( a{e}^{2}+c{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13421, size = 215, normalized size = 1.84 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \,{\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 = 1.80798, size = 328, normalized size = 2.8 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \,{\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 = 3.05701, size = 170, normalized size = 1.45 \begin{align*} - \frac{5 a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 20 c^{2} d e^{3} x^{3} + 15 c^{2} e^{4} x^{4} + x^{2} \left (15 a c e^{4} + 15 c^{2} d^{2} e^{2}\right ) + x \left (6 a c d e^{3} + 6 c^{2} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29139, size = 131, normalized size = 1.12 \begin{align*} -\frac{{\left (15 \, c^{2} x^{4} e^{4} + 20 \, c^{2} d x^{3} e^{3} + 15 \, c^{2} d^{2} x^{2} e^{2} + 6 \, c^{2} d^{3} x e + c^{2} d^{4} + 15 \, a c x^{2} e^{4} + 6 \, a c d x e^{3} + a c d^{2} e^{2} + 5 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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