Optimal. Leaf size=48 \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0141593, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 364} \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 364
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^n}}{\sqrt{1+x^n}} \, dx &=\left (x^{-n/2} \sqrt{a x^n}\right ) \int \frac{x^{n/2}}{\sqrt{1+x^n}} \, dx\\ &=\frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{2+n}\\ \end{align*}
Mathematica [A] time = 0.0105009, size = 40, normalized size = 0.83 \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-x^n\right )}{n+2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 35, normalized size = 0.7 \begin{align*} 2\,{\frac{x{\mbox{$_2$F$_1$}(1/2,1/2+{n}^{-1};\,3/2+{n}^{-1};\,-{x}^{n})}\sqrt{a{x}^{n}}}{2+n}} \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{\sqrt{a x^{n}}}{\sqrt{x^{n} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{n}}}{\sqrt{x^{n} + 1}}\, dx \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{\sqrt{a x^{n}}}{\sqrt{x^{n} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]