Optimal. Leaf size=181 \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.18466, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {866, 794, 648} \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 866
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} (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}} \, dx &=-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(4 g) \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}{c d}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e}-\frac{\left (4 g \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}{3 c^2 d^2 e}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{8 g \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}}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e}\\ \end{align*}
Mathematica [A] time = 0.07382, size = 88, normalized size = 0.49 \[ \frac{2 \sqrt{d+e x} \left (-8 a^2 e^2 g^2-4 a c d e g (g x-3 f)+c^2 d^2 \left (-3 f^2+6 f g x+g^2 x^2\right )\right )}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 116, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -{g}^{2}{x}^{2}{c}^{2}{d}^{2}+4\,acde{g}^{2}x-6\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-12\,acdefg+3\,{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17767, size = 132, normalized size = 0.73 \begin{align*} -\frac{2 \, f^{2}}{\sqrt{c d x + a e} c d} + \frac{4 \,{\left (c d x + 2 \, a e\right )} f g}{\sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} g^{2}}{3 \, \sqrt{c d x + a e} c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59079, size = 308, normalized size = 1.7 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} g^{2} x^{2} - 3 \, c^{2} d^{2} f^{2} + 12 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} + 2 \,{\left (3 \, c^{2} d^{2} f g - 2 \, a c d e g^{2}\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}}{3 \,{\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e +{\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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