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 = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = 0.01, 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]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{n}}}{\sqrt {x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 35, normalized size = 0.73 \[ \frac {2 \sqrt {a \,x^{n}}\, x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{n}+\frac {1}{2}\right ], \left [\frac {1}{n}+\frac {3}{2}\right ], -x^{n}\right )}{n +2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{n}}}{\sqrt {x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a\,x^n}}{\sqrt {x^n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x^{n}}}{\sqrt {x^{n} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________