Optimal. Leaf size=269 \[ \frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}-\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}} \]
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Rubi [A] time = 0.407257, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}-\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx &=\frac{2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac{(6 (c d f-a e g)) \int \frac{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{11 c d}\\ &=\frac{4 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac{\left (8 (c d f-a e g)^2\right ) \int \frac{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{33 c^2 d^2}\\ &=\frac{16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac{4 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac{\left (8 (c d f-a e g)^2 \left (7 f-\frac{5 d g}{e}-\frac{2 a e g}{c d}\right )\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)^{3/2}} \, dx}{231 c^2 d^2}\\ &=\frac{16 (c d f-a e g)^2 \left (7 f-\frac{5 d g}{e}-\frac{2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 c^3 d^3 (d+e x)^{5/2}}+\frac{16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac{4 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.179156, size = 137, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 c d e^2 g^2 (11 f+5 g x)-16 a^3 e^3 g^3-2 a c^2 d^2 e g \left (99 f^2+110 f g x+35 g^2 x^2\right )+c^3 d^3 \left (495 f^2 g x+231 f^3+385 f g^2 x^2+105 g^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 188, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -105\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+70\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-385\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-40\,{a}^{2}cd{e}^{2}{g}^{3}x+220\,a{c}^{2}{d}^{2}ef{g}^{2}x-495\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-88\,{a}^{2}cd{e}^{2}f{g}^{2}+198\,a{c}^{2}{d}^{2}e{f}^{2}g-231\,{f}^{3}{c}^{3}{d}^{3} \right ) }{1155\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17583, size = 397, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{3}}{5 \, c d} + \frac{6 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{2} g}{35 \, c^{2} d^{2}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f g^{2}}{105 \, c^{3} d^{3}} + \frac{2 \,{\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} e x^{4} + 5 \, a^{2} c^{3} d^{3} e^{2} x^{3} - 6 \, a^{3} c^{2} d^{2} e^{3} x^{2} + 8 \, a^{4} c d e^{4} x - 16 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} g^{3}}{1155 \, c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63672, size = 711, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (105 \, c^{5} d^{5} g^{3} x^{5} + 231 \, a^{2} c^{3} d^{3} e^{2} f^{3} - 198 \, a^{3} c^{2} d^{2} e^{3} f^{2} g + 88 \, a^{4} c d e^{4} f g^{2} - 16 \, a^{5} e^{5} g^{3} + 35 \,{\left (11 \, c^{5} d^{5} f g^{2} + 4 \, a c^{4} d^{4} e g^{3}\right )} x^{4} + 5 \,{\left (99 \, c^{5} d^{5} f^{2} g + 110 \, a c^{4} d^{4} e f g^{2} + a^{2} c^{3} d^{3} e^{2} g^{3}\right )} x^{3} + 3 \,{\left (77 \, c^{5} d^{5} f^{3} + 264 \, a c^{4} d^{4} e f^{2} g + 11 \, a^{2} c^{3} d^{3} e^{2} f g^{2} - 2 \, a^{3} c^{2} d^{2} e^{3} g^{3}\right )} x^{2} +{\left (462 \, a c^{4} d^{4} e f^{3} + 99 \, a^{2} c^{3} d^{3} e^{2} f^{2} g - 44 \, a^{3} c^{2} d^{2} e^{3} f g^{2} + 8 \, a^{4} c d e^{4} g^{3}\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}}{1155 \,{\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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