Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.12055, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )} \]
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)^{10}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{(6 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)^9} \, dx}{13 \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}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac{\left (24 c^2 d^2\right ) \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}{143 \left (c d^2-a e^2\right )^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac{\left (16 c^3 d^3\right ) \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}{429 \left (c d^2-a e^2\right )^3}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac{32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.0797922, size = 148, normalized size = 0.64 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (63 a^2 c d e^4 (13 d+2 e x)-231 a^3 e^6-7 a c^2 d^2 e^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )+c^3 d^3 \left (286 d^2 e x+429 d^3+104 d e^2 x^2+16 e^3 x^3\right )\right )}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 217, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+56\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-104\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-126\,{a}^{2}cd{e}^{5}x+364\,a{c}^{2}{d}^{3}{e}^{3}x-286\,{c}^{3}{d}^{5}ex+231\,{a}^{3}{e}^{6}-819\,{a}^{2}c{d}^{2}{e}^{4}+1001\,a{c}^{2}{d}^{4}{e}^{2}-429\,{c}^{3}{d}^{6} \right ) }{3003\, \left ( ex+d \right ) ^{9} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \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 [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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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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