Optimal. Leaf size=412 \[ -\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}+\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]
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Rubi [A] time = 0.627098, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \[ -\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}+\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 880
Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} (f+g x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{1}{9} \left (-9 d+\frac{8 a e^2}{c d}+\frac{e f}{g}\right ) \int \frac{\sqrt{d+e x} (f+g x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{\left (2 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )\right ) \int \frac{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{21 c^2 d^2 g}\\ &=-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )\right ) \int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{105 c^3 d^3 g}\\ &=-\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}+\frac{\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{315 c^4 d^4 e g}\\ &=\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^5 d^5 e g \sqrt{d+e x}}-\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.278514, size = 264, normalized size = 0.64 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-16 a^3 c d e^3 g^2 (9 d g+27 e f+4 e g x)+128 a^4 e^5 g^3-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (126 f^2 g x+105 f^3+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^2 g x+35 f^3+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (189 f^2 g x+105 f^3+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 425, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,e{g}^{3}{x}^{4}{c}^{4}{d}^{4}-40\,a{c}^{3}{d}^{3}{e}^{2}{g}^{3}{x}^{3}+45\,{c}^{4}{d}^{5}{g}^{3}{x}^{3}+135\,{c}^{4}{d}^{4}ef{g}^{2}{x}^{3}+48\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{g}^{3}{x}^{2}-54\,a{c}^{3}{d}^{4}e{g}^{3}{x}^{2}-162\,a{c}^{3}{d}^{3}{e}^{2}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{5}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{4}e{f}^{2}g{x}^{2}-64\,{a}^{3}cd{e}^{4}{g}^{3}x+72\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}{g}^{3}x+216\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}f{g}^{2}x-252\,a{c}^{3}{d}^{4}ef{g}^{2}x-252\,a{c}^{3}{d}^{3}{e}^{2}{f}^{2}gx+315\,{c}^{4}{d}^{5}{f}^{2}gx+105\,{c}^{4}{d}^{4}e{f}^{3}x+128\,{a}^{4}{e}^{5}{g}^{3}-144\,{a}^{3}c{d}^{2}{e}^{3}{g}^{3}-432\,{a}^{3}cd{e}^{4}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{f}^{2}g-630\,a{c}^{3}{d}^{4}e{f}^{2}g-210\,a{c}^{3}{d}^{3}{e}^{2}{f}^{3}+315\,{d}^{5}{f}^{3}{c}^{4} \right ) }{315\,{c}^{5}{d}^{5}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67286, size = 653, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} + \frac{2 \,{\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \,{\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} -{\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \,{\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \,{\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt{c d x + a e} c^{5} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53569, size = 837, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (35 \, c^{4} d^{4} e g^{3} x^{4} + 105 \,{\left (3 \, c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{3} - 126 \,{\left (5 \, a c^{3} d^{4} e - 4 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g + 72 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} f g^{2} - 16 \,{\left (9 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5}\right )} g^{3} + 5 \,{\left (27 \, c^{4} d^{4} e f g^{2} +{\left (9 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} g^{3}\right )} x^{3} + 3 \,{\left (63 \, c^{4} d^{4} e f^{2} g + 9 \,{\left (7 \, c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f g^{2} - 2 \,{\left (9 \, a c^{3} d^{4} e - 8 \, a^{2} c^{2} d^{2} e^{3}\right )} g^{3}\right )} x^{2} +{\left (105 \, c^{4} d^{4} e f^{3} + 63 \,{\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{2} g - 36 \,{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f g^{2} + 8 \,{\left (9 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} 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}}{315 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{3}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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