Optimal. Leaf size=93 \[ \frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \]
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Rubi [A] time = 0.529009, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2125, 2121, 12, 30} \[ \frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \]
Antiderivative was successfully verified.
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Rule 2125
Rule 2121
Rule 12
Rule 30
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}{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \, dx &=\frac{\sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \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}{\sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}} \, dx}{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}\\ &=\frac{\left (2 f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+n}}{2 e} \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )}{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}\\ &=\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right ) \operatorname{Subst}\left (\int x^{-1+n} \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )}{e \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}\\ &=\frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}}\\ \end{align*}
Mathematica [A] time = 0.0976617, size = 76, normalized size = 0.82 \[ \frac{f \sqrt{a+\frac{e x (2 d+e x)}{f^2}} \left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^n}{e n \sqrt{g \left (a+\frac{e x (2 d+e x)}{f^2}\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.071, 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}{\frac{1}{\sqrt{ag+2\,{\frac{degx}{{f}^{2}}}+{\frac{{e}^{2}g{x}^{2}}{{f}^{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}}{\sqrt{\frac{e^{2} g x^{2}}{f^{2}} + a g + \frac{2 \, d e g x}{f^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04327, size = 250, normalized size = 2.69 \begin{align*} \frac{{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f^{3} \sqrt{\frac{e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}}{e^{3} g n x^{2} + a e f^{2} g n + 2 \, d e^{2} g n x} \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}}{\sqrt{\frac{e^{2} g x^{2}}{f^{2}} + a g + \frac{2 \, d e g x}{f^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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