Optimal. Leaf size=121 \[ \frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
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Rubi [A] time = 0.10316, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2117, 947, 64} \[ \frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
Antiderivative was successfully verified.
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Rule 2117
Rule 947
Rule 64
Rubi steps
\begin{align*} \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^n \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \left (x^n+\frac{a f^2 x^n}{(d-x)^2}\right ) \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac{\left (a f^2\right ) \operatorname{Subst}\left (\int \frac{x^n}{(d-x)^2} \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac{a f^2 \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{1+n} \, _2F_1\left (2,1+n;2+n;\frac{d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}}{d}\right )}{2 d^2 e (1+n)}\\ \end{align*}
Mathematica [A] time = 0.117929, size = 86, normalized size = 0.71 \[ \frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \left (a f^2 \, _2F_1\left (2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right )+d^2\right )}{2 d^2 e (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, 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{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{n}\,{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 ({\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx \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{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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