Integrand size = 29, antiderivative size = 29 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\frac {d e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {d e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+d \text {Int}\left (\frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}}-\frac {c^2 d x (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}}\right ) \, dx \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\left (c^2 d\right ) \int \frac {x (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}} \, dx \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \sin \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (i d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}}-\frac {\left (i d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}} \\ & = \frac {d e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {d e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int \frac {\sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Not integrable
Time = 2.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}}{x}\, dx \]
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Not integrable
Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}}{x} \,d x \]
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