Optimal. Leaf size=48 \[ -\frac{2 (d+e x)^{3/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0215449, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {648} \[ -\frac{2 (d+e x)^{3/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\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}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0312852, size = 37, normalized size = 0.77 \[ -\frac{2 (d+e x)^{3/2}}{3 c d ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 50, normalized size = 1. \begin{align*} -{\frac{2\,cdx+2\,ae}{3\,cd} \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 = 1.09454, size = 38, normalized size = 0.79 \begin{align*} -\frac{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 [B] time = 1.50973, size = 221, normalized size = 4.6 \begin{align*} -\frac{2 \, \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^{3} d^{3} e x^{3} + a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (2 \, a c^{2} d^{3} e + a^{2} c d 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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