Optimal. Leaf size=104 \[ -\frac{(f+g x)^{n+1} (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, _2F_1\left (1,n+\frac{9}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{(n+1) (d+e x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.111715, 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} \[ \frac{2 (f+g x)^n (a e+c d x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{7 c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 891
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(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}}{(d+e x)^{5/2}} \, dx &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \int (a e+c d x)^{5/2} (f+g x)^n \, dx}{\sqrt{a e+c d x} \sqrt{d+e x}}\\ &=\frac{\left ((f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}\right ) \int (a e+c d x)^{5/2} \left (\frac{c d f}{c d f-a e g}+\frac{c d g x}{c d f-a e g}\right )^n \, dx}{\sqrt{a e+c d x} \sqrt{d+e x}}\\ &=\frac{2 (a e+c d x)^3 (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{7 c d \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0723055, size = 110, normalized size = 1.06 \[ \frac{2 (f+g x)^n (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{7 c d \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.705, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{n} \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{n}}{{\left (e x + d\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{{\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}{\left (g x + f\right )}^{n}}{\sqrt{e x + d}}, 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{5}{2}}{\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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