Integrand size = 25, antiderivative size = 25 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \text {Int}\left (\frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2},x\right )}{2 d} \]
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Not integrable
Time = 0.10 (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 = {} \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {x^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {(1-m) \int \frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx}{2 d} \\ & = \frac {x^{1+m} (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {b c x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{2 d^2 (2+m)}+\frac {(1-m) \int \frac {x^m (a+b \arcsin (c x))}{d-c^2 d x^2} \, dx}{2 d} \\ \end{align*}
Not integrable
Time = 5.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
\[\int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Not integrable
Time = 16.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{m} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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