Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 1.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
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Not integrable
Time = 2.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 140.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.99 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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