Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\text {Int}\left (\frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}},x\right ) \]
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Not integrable
Time = 0.03 (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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ \end{align*}
Not integrable
Time = 2.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
\[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\operatorname {arcsinh}\left (a x \right )}}d x\]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 58.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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Not integrable
Time = 0.50 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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Not integrable
Time = 2.72 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {asinh}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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