Integrand size = 29, antiderivative size = 29 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=-\frac {2 b c x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(2+m)^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2+m}+\frac {2 b^2 c^2 x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m)^2 (3+m) \sqrt {1-c^2 x^2}}+\frac {d \text {Int}\left (\frac {x^m (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{2+m} \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 29, 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 = {} \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2+m}+\frac {d \int \frac {x^m (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}-\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int x^{1+m} (a+b \arcsin (c x)) \, dx}{(2+m) \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b c x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(2+m)^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2+m}+\frac {d \int \frac {x^m (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m}+\frac {\left (2 b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^{2+m}}{\sqrt {1-c^2 x^2}} \, dx}{(2+m)^2 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b c x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(2+m)^2 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2+m}+\frac {2 b^2 c^2 x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m)^2 (3+m) \sqrt {1-c^2 x^2}}+\frac {d \int \frac {x^m (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{2+m} \\ \end{align*}
Not integrable
Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx \]
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Not integrable
Time = 0.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int x^{m} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]
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Not integrable
Time = 20.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^{m} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
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Not integrable
Time = 0.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \]
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Exception generated. \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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