Optimal. Leaf size=269 \[ \frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt{d+e x}}-\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.39, 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} \[ \frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt{d+e x}}-\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, 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 )^{3/2}}{9 c d (d+e x)^{3/2}}+\frac{\left (2 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{3 c d e^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 )^{3/2}}{21 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{9 c d (d+e x)^{3/2}}+\frac{\left (8 (c d f-a e g)^2\right ) \int \frac{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{21 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 )^{3/2}}{105 c^3 d^3 e \sqrt{d+e x}}+\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 )^{3/2}}{21 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{9 c d (d+e x)^{3/2}}+\frac{\left (8 (c d f-a e g)^2 \left (5 f-\frac{3 d g}{e}-\frac{2 a e g}{c d}\right )\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}{105 c^2 d^2}\\ &=\frac{16 (c d f-a e g)^2 \left (5 f-\frac{3 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 )^{3/2}}{315 c^3 d^3 (d+e x)^{3/2}}+\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 )^{3/2}}{105 c^3 d^3 e \sqrt{d+e x}}+\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 )^{3/2}}{21 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{9 c d (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.124367, size = 136, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (24 a^2 c d e^2 g^2 (3 f+g x)-16 a^3 e^3 g^3-6 a c^2 d^2 e g \left (21 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (189 f^2 g x+105 f^3+135 f g^2 x^2+35 g^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 188, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -35\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+30\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-135\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-24\,{a}^{2}cd{e}^{2}{g}^{3}x+108\,a{c}^{2}{d}^{2}ef{g}^{2}x-189\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-72\,{a}^{2}cd{e}^{2}f{g}^{2}+126\,a{c}^{2}{d}^{2}e{f}^{2}g-105\,{f}^{3}{c}^{3}{d}^{3} \right ) }{315\,{c}^{4}{d}^{4}}\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 = 1.23418, size = 294, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{3}}{3 \, c d} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{2} g}{5 \, c^{2} d^{2}} + \frac{2 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f g^{2}}{35 \, c^{3} d^{3}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} g^{3}}{315 \, c^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62027, size = 554, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (35 \, c^{4} d^{4} g^{3} x^{4} + 105 \, a c^{3} d^{3} e f^{3} - 126 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 72 \, a^{3} c d e^{3} f g^{2} - 16 \, a^{4} e^{4} g^{3} + 5 \,{\left (27 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} x^{3} + 3 \,{\left (63 \, c^{4} d^{4} f^{2} g + 9 \, a c^{3} d^{3} e f g^{2} - 2 \, a^{2} c^{2} d^{2} e^{2} g^{3}\right )} x^{2} +{\left (105 \, c^{4} d^{4} f^{3} + 63 \, a c^{3} d^{3} e f^{2} g - 36 \, a^{2} c^{2} d^{2} e^{2} f g^{2} + 8 \, a^{3} c d e^{3} 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^{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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