Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=-\frac {\left (1-c^2 x^2\right )^2}{b c x^2 (a+b \arcsin (c x))}-\frac {2 \text {Int}\left (\frac {1-c^2 x^2}{x^3 (a+b \arcsin (c x))},x\right )}{b c}-\frac {2 c \text {Int}\left (\frac {1-c^2 x^2}{x (a+b \arcsin (c x))},x\right )}{b} \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1-c^2 x^2\right )^2}{b c x^2 (a+b \arcsin (c x))}-\frac {2 \int \frac {1-c^2 x^2}{x^3 (a+b \arcsin (c x))} \, dx}{b c}-\frac {(2 c) \int \frac {1-c^2 x^2}{x (a+b \arcsin (c x))} \, dx}{b} \\ \end{align*}
Not integrable
Time = 3.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 2.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.93 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 0.80 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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