Integrand size = 14, antiderivative size = 179 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} d} \]
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Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4887, 4717, 4807, 4719, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4717
Rule 4719
Rule 4807
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d}-\frac {\left (4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (8 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d}-\frac {\left (8 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c+d x))}{b}} \left (-2 b e^{i \arcsin (c+d x)} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )-i e^{\frac {i a}{b}} \left (2 a \left (-1+e^{2 i \arcsin (c+d x)}\right )+b \left (-i-2 \arcsin (c+d x)+e^{2 i \arcsin (c+d x)} (-i+2 \arcsin (c+d x))\right )-2 i b e^{\frac {i (a+b \arcsin (c+d x))}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{3 b^2 d (a+b \arcsin (c+d x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(145)=290\).
Time = 0.31 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {2 \left (2 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b -2 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b +2 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, a -2 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, a +2 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b +\cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \right )}{3 d \,b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(370\) |
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Exception generated. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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