Optimal. Leaf size=233 \[ \frac{12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}} \]
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Rubi [A] time = 0.179461, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, 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 c d \sqrt{d+e x}}+\frac{\left (6 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx}{13 d}\\ &=\frac{12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}}+\frac{\left (24 \left (d^2-\frac{a e^2}{c}\right )^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)^{3/2}} \, dx}{143 d^2}\\ &=\frac{16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}}+\frac{\left (16 \left (d^2-\frac{a e^2}{c}\right )^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)^{5/2}} \, dx}{429 d^3}\\ &=\frac{32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.121546, size = 142, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 c d e^4 (13 d+7 e x)-16 a^3 e^6-2 a c^2 d^2 e^2 \left (143 d^2+182 d e x+63 e^2 x^2\right )+c^3 d^3 \left (1001 d^2 e x+429 d^3+819 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 168, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -231\,{e}^{3}{x}^{3}{c}^{3}{d}^{3}+126\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-819\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-56\,{a}^{2}cd{e}^{5}x+364\,a{c}^{2}{d}^{3}{e}^{3}x-1001\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-104\,{a}^{2}c{d}^{2}{e}^{4}+286\,a{c}^{2}{d}^{4}{e}^{2}-429\,{c}^{3}{d}^{6} \right ) }{3003\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13945, size = 452, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \,{\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \,{\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} +{\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{3003 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88676, size = 752, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \,{\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \,{\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} +{\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{3003 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\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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