Optimal. Leaf size=122 \[ -\frac{8 f^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+5}{2};\frac{\left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+3) \left (d^2-a f^2\right )^3} \]
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Rubi [A] time = 0.267613, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2121, 12, 364} \[ -\frac{8 f^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+5}{2};\frac{\left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+\frac{2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+3) \left (d^2-a f^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 2121
Rule 12
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n}{\left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right )^2} \, dx &=\left (2 f^4\right ) \operatorname{Subst}\left (\int \frac{4 e^2 x^{2+n}}{\left (d^2 e-\left (-a e+\frac{2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3} \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )\\ &=\left (8 e^2 f^4\right ) \operatorname{Subst}\left (\int \frac{x^{2+n}}{\left (d^2 e-\left (-a e+\frac{2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3} \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )\\ &=-\frac{8 f^4 \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^{3+n} \, _2F_1\left (3,\frac{3+n}{2};\frac{5+n}{2};\frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)}\\ \end{align*}
Mathematica [A] time = 0.193331, size = 112, normalized size = 0.92 \[ -\frac{8 f^4 \left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^{n+3} \, _2F_1\left (3,\frac{n+3}{2};\frac{n+5}{2};\frac{\left (d+e x+f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e (n+3) \left (d^2-a f^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n} \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{-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 + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}}{{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}^{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 (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + a^{2} f^{4} + 4 \, a d e f^{2} x + 2 \,{\left (a e^{2} f^{2} + 2 \, d^{2} e^{2}\right )} x^{2}}, 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 (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}}{{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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