Optimal. Leaf size=63 \[ -\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.0663038, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {860} \[ -\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2} \sqrt{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 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) \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 = 0.0334774, size = 52, normalized size = 0.83 \[ -\frac{2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 63, normalized size = 1. \begin{align*}{\frac{2\,cdx+2\,ae}{3\,aeg-3\,cdf} \left ( gx+f \right ) ^{{\frac{3}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}} \sqrt{g x + f}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72344, size = 394, normalized size = 6.25 \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}{\left (g x + f\right )}^{\frac{3}{2}}}{3 \,{\left (a^{2} c d^{2} e^{2} f - a^{3} d e^{3} g +{\left (c^{3} d^{3} e f - a c^{2} d^{2} e^{2} g\right )} x^{3} +{\left ({\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f -{\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} g\right )} x^{2} +{\left ({\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f -{\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g\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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