Optimal. Leaf size=74 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{m+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0419184, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6124, 808, 364} \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6124
Rule 808
Rule 364
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^m \, dx &=\int \frac{x^m (1+a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=a \int \frac{x^{1+m}}{\sqrt{1-a^2 x^2}} \, dx+\int \frac{x^m}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{1+m}+\frac{a x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{2+m}\\ \end{align*}
Mathematica [C] time = 0.0299778, size = 70, normalized size = 0.95 \[ -\frac{\sqrt{-a x-1} \sqrt{1-a x} x^{m+1} F_1\left (m+1;-\frac{1}{2},\frac{1}{2};m+2;-a x,a x\right )}{(m+1) \sqrt{a x-1} \sqrt{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.234, size = 67, normalized size = 0.9 \begin{align*}{\frac{{x}^{1+m}}{1+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{3}{2}}+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}+{\frac{a{x}^{2+m}}{2+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+{\frac{m}{2}};\,2+{\frac{m}{2}};\,{a}^{2}{x}^{2})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{m}}{a x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 3.7963, size = 97, normalized size = 1.31 \begin{align*} \frac{a x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]