Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x \sqrt {\arcsin (a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\arcsin (a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\arcsin (a x)}},x\right )}{6 c^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\arcsin (a x)}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}} \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\arcsin (a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\arcsin (a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}} \\ & = \frac {x \sqrt {\arcsin (a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\arcsin (a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\arcsin (a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right ) \sqrt {\arcsin (a x)}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}} \\ \end{align*}
Not integrable
Time = 2.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int \frac {\sqrt {\arcsin \left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 22.75 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\operatorname {asin}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 0.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\arcsin \left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\mathrm {asin}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
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