Optimal. Leaf size=154 \[ \frac{2 \sqrt{d+e x} \left (2 a e^2 g+c d (e f-3 d g)\right )}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{5/2} (c d f-a e g)}{3 c d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.133725, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {788, 648} \[ \frac{2 \sqrt{d+e x} \left (2 a e^2 g+c d (e f-3 d g)\right )}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{5/2} (c d f-a e g)}{3 c d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 (c d f-a e g) (d+e x)^{5/2}}{3 c d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (2 a e^2 g+c d (e f-3 d g)\right ) \int \frac{(d+e x)^{3/2}}{\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 \left (c d^2-a e^2\right )}\\ &=-\frac{2 (c d f-a e g) (d+e x)^{5/2}}{3 c d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{2 \left (2 a e^2 g+c d (e f-3 d g)\right ) \sqrt{d+e x}}{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}}\\ \end{align*}
Mathematica [A] time = 0.0498132, size = 52, normalized size = 0.34 \[ -\frac{2 (d+e x)^{3/2} (2 a e g+c d (f+3 g x))}{3 c^2 d^2 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 66, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,xcdg+2\,aeg+cdf \right ) }{3\,{c}^{2}{d}^{2}} \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 = 1.17725, size = 99, normalized size = 0.64 \begin{align*} -\frac{2 \,{\left (3 \, c d x + 2 \, a e\right )} g}{3 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt{c d x + a e}} - \frac{2 \, f}{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 = 1.56673, size = 270, normalized size = 1.75 \begin{align*} -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (3 \, c d g x + c d f + 2 \, a e g\right )} \sqrt{e x + d}}{3 \,{\left (c^{4} d^{4} e x^{3} + a^{2} c^{2} d^{3} e^{2} +{\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} x^{2} +{\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} 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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