Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]
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Not integrable
Time = 0.08 (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 = {} \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 19.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
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Not integrable
Time = 0.81 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {x^{4} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 40.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 4.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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