Optimal. Leaf size=211 \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{3 c^3 d^3 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{8 g (d+e x)^{3/2} (c d f-a e g)}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.219514, antiderivative size = 211, 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, 788, 648} \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{3 c^3 d^3 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{8 g (d+e x)^{3/2} (c d f-a e g)}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 866
Rule 788
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2} (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}} \, dx &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(4 g) \int \frac{(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 g (c d f-a e g) (d+e x)^{3/2}}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (4 g \left (2 a e^2 g-c d (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 \left (c d^2-a e^2\right )}\\ &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 g (c d f-a e g) (d+e x)^{3/2}}{3 c^2 d^2 \left (c d^2-a e^2\right ) \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 (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 \left (c d^2-a e^2\right ) \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0746126, size = 87, normalized size = 0.41 \[ \frac{2 (d+e x)^{3/2} \left (8 a^2 e^2 g^2-4 a c d e g (f-3 g x)-c^2 d^2 \left (f^2+6 f g x-3 g^2 x^2\right )\right )}{3 c^3 d^3 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 116, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}+12\,acde{g}^{2}x-6\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-4\,acdefg-{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24443, size = 186, normalized size = 0.88 \begin{align*} -\frac{4 \,{\left (3 \, c d x + 2 \, a e\right )} f g}{3 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt{c d x + a e}} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d e x + 8 \, a^{2} e^{2}\right )} g^{2}}{3 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt{c d x + a e}} - \frac{2 \, f^{2}}{3 \,{\left (c^{2} d^{2} x + a c d e\right )} \sqrt{c d x + a e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55553, size = 366, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} g^{2} x^{2} - c^{2} d^{2} f^{2} - 4 \, a c d e f g + 8 \, a^{2} e^{2} g^{2} - 6 \,{\left (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^{5} d^{5} e x^{3} + a^{2} c^{3} d^{4} e^{2} +{\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} x^{2} +{\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\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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