Integrand size = 12, antiderivative size = 69 \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \arcsin \left (c x^n\right )\right )-\frac {b c n x^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2+n)} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4926, 12, 371} \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \arcsin \left (c x^n\right )\right )-\frac {b c n x^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (n+2)} \]
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Rule 12
Rule 371
Rule 4926
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \left (a+b \arcsin \left (c x^n\right )\right )-\frac {1}{2} b \int \frac {c n x^{1+n}}{\sqrt {1-c^2 x^{2 n}}} \, dx \\ & = \frac {1}{2} x^2 \left (a+b \arcsin \left (c x^n\right )\right )-\frac {1}{2} (b c n) \int \frac {x^{1+n}}{\sqrt {1-c^2 x^{2 n}}} \, dx \\ & = \frac {1}{2} x^2 \left (a+b \arcsin \left (c x^n\right )\right )-\frac {b c n x^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\frac {a x^2}{2}+\frac {1}{2} b x^2 \arcsin \left (c x^n\right )-\frac {b c n x^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2 n},1+\frac {2+n}{2 n},c^2 x^{2 n}\right )}{2 (2+n)} \]
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\[\int x \left (a +b \arcsin \left (c \,x^{n}\right )\right )d x\]
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Exception generated. \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Result contains complex when optimal does not.
Time = 2.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\frac {a x^{2}}{2} + \frac {i b c c^{\frac {2}{n}} c^{-1 - \frac {2}{n}} x^{2} \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - \frac {1}{n} \\ 1 - \frac {1}{n} \end {matrix}\middle | {\frac {x^{- 2 n}}{c^{2}}} \right )}}{4 \Gamma \left (1 + \frac {1}{n}\right )} + \frac {b x^{2} \operatorname {asin}{\left (c x^{n} \right )}}{2} \]
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\[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\int { {\left (b \arcsin \left (c x^{n}\right ) + a\right )} x \,d x } \]
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\[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\int { {\left (b \arcsin \left (c x^{n}\right ) + a\right )} x \,d x } \]
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Timed out. \[ \int x \left (a+b \arcsin \left (c x^n\right )\right ) \, dx=\int x\,\left (a+b\,\mathrm {asin}\left (c\,x^n\right )\right ) \,d x \]
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