Optimal. Leaf size=63 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.0702738, 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} \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0542479, size = 52, normalized size = 0.83 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 63, normalized size = 1. \begin{align*} -{\frac{2\,cdx+2\,ae}{5\,aeg-5\,cdf} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47522, size = 482, normalized size = 7.65 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{5 \,{\left (c d^{2} f^{4} - a d e f^{3} g +{\left (c d e f g^{3} - a e^{2} g^{4}\right )} x^{4} +{\left (3 \, c d e f^{2} g^{2} - a d e g^{4} +{\left (c d^{2} - 3 \, a e^{2}\right )} f g^{3}\right )} x^{3} + 3 \,{\left (c d e f^{3} g - a d e f g^{3} +{\left (c d^{2} - a e^{2}\right )} f^{2} g^{2}\right )} x^{2} +{\left (c d e f^{4} - 3 \, a d e f^{2} g^{2} +{\left (3 \, c d^{2} - a e^{2}\right )} f^{3} 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*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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