Integrand size = 18, antiderivative size = 107 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arcsin (c x))^2}{d}-\frac {8 b c (d x)^{3/2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )}{3 d^2}+\frac {16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )}{15 d^3} \]
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Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4723, 4805} \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\frac {16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )}{15 d^3}-\frac {8 b c (d x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right ) (a+b \arcsin (c x))}{3 d^2}+\frac {2 \sqrt {d x} (a+b \arcsin (c x))^2}{d} \]
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Rule 4723
Rule 4805
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d x} (a+b \arcsin (c x))^2}{d}-\frac {(4 b c) \int \frac {\sqrt {d x} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{d} \\ & = \frac {2 \sqrt {d x} (a+b \arcsin (c x))^2}{d}-\frac {8 b c (d x)^{3/2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )}{3 d^2}+\frac {16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )}{15 d^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\frac {2 x \left (5 (a+b \arcsin (c x)) \left (3 (a+b \arcsin (c x))-4 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )+8 b^2 c^2 x^2 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )\right )}{15 \sqrt {d x}} \]
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\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {d x}}d x\]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {d x}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d\,x}} \,d x \]
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