Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {x \sqrt {\text {arcsinh}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arcsinh}(a x)}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\text {arcsinh}(a x)}},x\right )}{6 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x}{\left (1+a^2 x^2\right ) \sqrt {\text {arcsinh}(a x)}},x\right )}{3 c^2 \sqrt {c+a^2 c x^2}} \]
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Not integrable
Time = 0.13 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\text {arcsinh}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\text {arcsinh}(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 {\text {arcsinh}(a x)}} \, dx}{6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {\text {arcsinh}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {arcsinh}(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 {\text {arcsinh}(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 {\text {arcsinh}(a x)}} \, dx}{3 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Not integrable
Time = 2.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
\[\int \frac {\sqrt {\operatorname {arcsinh}\left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 23.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\operatorname {asinh}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 2.80 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\mathrm {asinh}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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