Optimal. Leaf size=81 \[ \frac{x^{m+1} \left (a+b \sin ^{-1}\left (c x^n\right )\right )}{m+1}-\frac{b c n x^{m+n+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},c^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0513925, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 364} \[ \frac{x^{m+1} \left (a+b \sin ^{-1}\left (c x^n\right )\right )}{m+1}-\frac{b c n x^{m+n+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4842
Rule 12
Rule 364
Rubi steps
\begin{align*} \int x^m \left (a+b \sin ^{-1}\left (c x^n\right )\right ) \, dx &=\frac{x^{1+m} \left (a+b \sin ^{-1}\left (c x^n\right )\right )}{1+m}-\frac{b \int \frac{c n x^{m+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}\left (c x^n\right )\right )}{1+m}-\frac{(b c n) \int \frac{x^{m+n}}{\sqrt{1-c^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}\left (c x^n\right )\right )}{1+m}-\frac{b c n x^{1+m+n} \, _2F_1\left (\frac{1}{2},\frac{1+m+n}{2 n};\frac{1+m+3 n}{2 n};c^2 x^{2 n}\right )}{(1+m) (1+m+n)}\\ \end{align*}
Mathematica [A] time = 0.110121, size = 78, normalized size = 0.96 \[ \frac{x^{m+1} \left ((m+n+1) \left (a+b \sin ^{-1}\left (c x^n\right )\right )-b c n x^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},c^2 x^{2 n}\right )\right )}{(m+1) (m+n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b\arcsin \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 x^{m} \left (a + b \operatorname{asin}{\left (c x^{n} \right )}\right )\, 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{\left (b \arcsin \left (c x^{n}\right ) + a\right )} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]