Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Int}\left (\frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)},x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx \\ \end{align*}
Not integrable
Time = 0.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {x^{m} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{a +b \,\operatorname {arcsinh}\left (c x \right )}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Not integrable
Time = 14.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^{m} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Exception generated. \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (1+c^2 x^2\right )^{3/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x^m\,{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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