Integrand size = 23, antiderivative size = 23 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\text {Int}\left (\frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}},x\right ) \]
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Not integrable
Time = 0.10 (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 = {} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \\ \end{align*}
Not integrable
Time = 15.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]
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Not integrable
Time = 2.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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Not integrable
Time = 1.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^3\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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