Integrand size = 16, antiderivative size = 261 \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=-\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arcsin \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arcsin \left (1+d x^2\right )}}+\frac {\sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )}+\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )} \]
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Time = 0.04 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4912, 4903} \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {\sqrt {\pi } x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arcsin \left (d x^2+1\right )}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (d x^2+1\right )\right )\right )}+\frac {\sqrt {\pi } x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arcsin \left (d x^2+1\right )}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (d x^2+1\right )\right )\right )}+\frac {x}{3 b^2 \sqrt {a+b \arcsin \left (d x^2+1\right )}}-\frac {\sqrt {-d^2 x^4-2 d x^2}}{3 b d x \left (a+b \arcsin \left (d x^2+1\right )\right )^{3/2}} \]
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Rule 4903
Rule 4912
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arcsin \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arcsin \left (1+d x^2\right )}}-\frac {\int \frac {1}{\sqrt {a+b \arcsin \left (1+d x^2\right )}} \, dx}{3 b^2} \\ & = -\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arcsin \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arcsin \left (1+d x^2\right )}}+\frac {\sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )}+\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {x \left (\frac {b \left (2+d x^2\right )}{\sqrt {-d x^2 \left (2+d x^2\right )} \left (a+b \arcsin \left (1+d x^2\right )\right )^{3/2}}+\frac {1}{\sqrt {a+b \arcsin \left (1+d x^2\right )}}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arcsin \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {b} \left (\cos \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1+d x^2\right )\right )\right )}\right )}{3 b^2} \]
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\[\int \frac {1}{{\left (a +b \arcsin \left (d \,x^{2}+1\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \arcsin \left (1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (d\,x^2+1\right )\right )}^{5/2}} \,d x \]
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