Optimal. Leaf size=104 \[ -\frac{(d+e x)^{5/2} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left (1,n-\frac{1}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{(n+1) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.119212, antiderivative size = 122, normalized size of antiderivative = 1.17, 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} \[ -\frac{2 \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{3 c d (a e+c d x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 891
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2} (f+g x)^n}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{a e+c d x} \sqrt{d+e x}\right ) \int \frac{(f+g x)^n}{(a e+c d x)^{5/2}} \, dx}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{\left (\sqrt{a e+c d x} \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n}\right ) \int \frac{\left (\frac{c d f}{c d f-a e g}+\frac{c d g x}{c d f-a e g}\right )^n}{(a e+c d x)^{5/2}} \, dx}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{3 c d (a e+c d x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.085808, size = 100, normalized size = 0.96 \[ -\frac{2 (d+e x)^{3/2} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{3 c d ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.763, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{n} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{-{\frac{5}{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 (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{n}}{{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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 )}^{n}}{c^{3} d^{3} e x^{4} + a^{3} d e^{3} +{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{3} + 3 \,{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x}, 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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