Integrand size = 18, antiderivative size = 109 \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c x))^2}{3 d (d x)^{3/2}}-\frac {8 b c (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^2 x^2\right )}{3 d^2 \sqrt {d x}}+\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3} \]
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Time = 0.09 (sec) , antiderivative size = 109, 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} \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^2 x^2\right ) (a+b \arcsin (c x))}{3 d^2 \sqrt {d x}}-\frac {2 (a+b \arcsin (c x))^2}{3 d (d x)^{3/2}} \]
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Rule 4723
Rule 4805
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b \arcsin (c x))^2}{3 d (d x)^{3/2}}+\frac {(4 b c) \int \frac {a+b \arcsin (c x)}{(d x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 d} \\ & = -\frac {2 (a+b \arcsin (c x))^2}{3 d (d x)^{3/2}}-\frac {8 b c (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^2 x^2\right )}{3 d^2 \sqrt {d x}}+\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\frac {x \left (-2 (a+b \arcsin (c x)) \left (a+b \arcsin (c x)+4 b c x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^2 x^2\right )\right )+16 b^2 c^2 x^2 \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )\right )}{3 (d x)^{5/2}} \]
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\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]
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