Optimal. Leaf size=231 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.115717, 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} \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \]
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 )^{3/2}}{(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 )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\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 )^{3/2}}{(d+e x)^7} \, dx}{11 \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 )^{5/2}}{11 \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 )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac{\left (8 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 )^{3/2}}{(d+e x)^6} \, dx}{33 \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 )^{5/2}}{11 \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 )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\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 )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\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 )^{3/2}}{(d+e x)^5} \, dx}{231 \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 )^{5/2}}{11 \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 )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\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 )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\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 )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5}\\ \end{align*}
Mathematica [A] time = 0.0909509, size = 138, normalized size = 0.6 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 c d e^4 (11 d+2 e x)-105 a^3 e^6-5 a c^2 d^2 e^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )+c^3 d^3 \left (198 d^2 e x+231 d^3+88 d e^2 x^2+16 e^3 x^3\right )\right )}{1155 (d+e x)^8 \left (c d^2-a e^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, 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}+40\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-88\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-70\,{a}^{2}cd{e}^{5}x+220\,a{c}^{2}{d}^{3}{e}^{3}x-198\,{c}^{3}{d}^{5}ex+105\,{a}^{3}{e}^{6}-385\,{a}^{2}c{d}^{2}{e}^{4}+495\,a{c}^{2}{d}^{4}{e}^{2}-231\,{c}^{3}{d}^{6} \right ) }{1155\, \left ( ex+d \right ) ^{7} \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{3}{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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