Optimal. Leaf size=105 \[ \frac{2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
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Rubi [A] time = 0.080604, antiderivative size = 105, 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 = {891, 70, 69} \[ \frac{2 (f+g x)^{5/2} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
Antiderivative was successfully verified.
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Rule 891
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (a e+c d x)^{-m} (f+g x)^{3/2} \, dx\\ &=\left (\left (\frac{g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int (f+g x)^{3/2} \left (-\frac{a e g}{c d f-a e g}-\frac{c d g x}{c d f-a e g}\right )^{-m} \, dx\\ &=\frac{2 \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g}\\ \end{align*}
Mathematica [A] time = 0.058219, size = 93, normalized size = 0.89 \[ \frac{2 (f+g x)^{5/2} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left (\frac{g (a e+c d x)}{a e g-c d f}\right )^m \, _2F_1\left (\frac{5}{2},m;\frac{7}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{5 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.638, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}\, dx \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 (g x + f\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (g x + f\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}, 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 (g x + f\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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