Optimal. Leaf size=111 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0488611, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac{(2 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.0408023, size = 71, normalized size = 0.64 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (c d (9 d+2 e x)-7 a e^2\right )}{63 (d+e x)^5 \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 90, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,cdex+7\,a{e}^{2}-9\,c{d}^{2} \right ) }{63\, \left ( ex+d \right ) ^{7} \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 96.0911, size = 686, normalized size = 6.18 \begin{align*} \frac{2 \,{\left (2 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 7 \, a^{4} e^{5} +{\left (9 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \,{\left (9 \, a c^{3} d^{4} e - 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} +{\left (27 \, a^{2} c^{2} d^{3} e^{2} - 19 \, a^{3} c d e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{63 \,{\left (c^{2} d^{9} - 2 \, a c d^{7} e^{2} + a^{2} d^{5} e^{4} +{\left (c^{2} d^{4} e^{5} - 2 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} x^{5} + 5 \,{\left (c^{2} d^{5} e^{4} - 2 \, a c d^{3} e^{6} + a^{2} d e^{8}\right )} x^{4} + 10 \,{\left (c^{2} d^{6} e^{3} - 2 \, a c d^{4} e^{5} + a^{2} d^{2} e^{7}\right )} x^{3} + 10 \,{\left (c^{2} d^{7} e^{2} - 2 \, a c d^{5} e^{4} + a^{2} d^{3} e^{6}\right )} x^{2} + 5 \,{\left (c^{2} d^{8} e - 2 \, a c d^{6} e^{3} + a^{2} d^{4} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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