Optimal. Leaf size=295 \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]
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Rubi [A] time = 0.262823, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int (d+e x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac{2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac{\left (8 \left (d^2-\frac{a e^2}{c}\right )\right ) \int (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{11 d}\\ &=\frac{16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac{\left (16 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{33 d^2}\\ &=\frac{32 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac{\left (64 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{231 d^3}\\ &=\frac{128 \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 )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac{\left (128 \left (d^2-\frac{a e^2}{c}\right )^4\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{1155 d^4}\\ &=\frac{256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac{128 \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 )^{3/2}}{1155 c^4 d^4 \sqrt{d+e x}}+\frac{32 \left (c d^2-a e^2\right )^2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac{16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac{2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}\\ \end{align*}
Mathematica [A] time = 0.157107, size = 187, normalized size = 0.63 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (48 a^2 c^2 d^2 e^4 \left (33 d^2+22 d e x+5 e^2 x^2\right )-64 a^3 c d e^6 (11 d+3 e x)+128 a^4 e^8-8 a c^3 d^3 e^2 \left (297 d^2 e x+231 d^3+165 d e^2 x^2+35 e^3 x^3\right )+c^4 d^4 \left (2970 d^2 e^2 x^2+2772 d^3 e x+1155 d^4+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 243, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 315\,{e}^{4}{x}^{4}{c}^{4}{d}^{4}-280\,a{c}^{3}{d}^{3}{e}^{5}{x}^{3}+1540\,{c}^{4}{d}^{5}{e}^{3}{x}^{3}+240\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{2}-1320\,a{c}^{3}{d}^{4}{e}^{4}{x}^{2}+2970\,{c}^{4}{d}^{6}{e}^{2}{x}^{2}-192\,{a}^{3}cd{e}^{7}x+1056\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}x-2376\,a{c}^{3}{d}^{5}{e}^{3}x+2772\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-704\,{a}^{3}c{d}^{2}{e}^{6}+1584\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-1848\,a{c}^{3}{d}^{6}{e}^{2}+1155\,{c}^{4}{d}^{8} \right ) }{3465\,{c}^{5}{d}^{5}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09053, size = 400, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \,{\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \,{\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{3465 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91222, size = 679, normalized size = 2.3 \begin{align*} \frac{2 \,{\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \,{\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \,{\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} +{\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} 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}}{3465 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\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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