Integrand size = 23, antiderivative size = 23 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Int}\left (x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]
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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 = {} \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \\ \end{align*}
Not integrable
Time = 12.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]
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Not integrable
Time = 1.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Timed out} \]
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Exception generated. \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 1.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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