Integrand size = 16, antiderivative size = 384 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {4 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d^2}+\frac {4 c \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 b^{5/2} d^2}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d^2} \]
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Time = 0.61 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {4889, 4829, 4717, 4807, 4719, 3387, 3386, 3432, 3385, 3433, 4729, 4731, 4491, 12, 4737} \[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {8 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d^2}+\frac {4 \sqrt {2 \pi } c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {4 \sqrt {2 \pi } c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d^2}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}} \]
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4717
Rule 4719
Rule 4729
Rule 4731
Rule 4737
Rule 4807
Rule 4829
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {c}{d (a+b \arcsin (x))^{5/2}}+\frac {x}{d (a+b \arcsin (x))^{5/2}}\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}-\frac {4 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}+\frac {(2 c) \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^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {16 \text {Subst}\left (\int \frac {x}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{3 b^2 d^2}+\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{3 b^2 d^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d^2}+\frac {(4 c) \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^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d^2}+\frac {\left (4 c \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^2}+\frac {\left (4 c \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^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d^2}+\frac {\left (8 c \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^2}+\frac {\left (8 c \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^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {4 c \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^2}+\frac {4 c \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^2}-\frac {\left (8 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d^2}+\frac {\left (8 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {4 c \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^2}+\frac {4 c \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^2}-\frac {\left (16 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d^2}+\frac {\left (16 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{3 b d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {4 c \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^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d^2}+\frac {4 c \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^2}+\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d^2} \\ \end{align*}
Time = 6.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.01 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {4 a \sqrt {b} c^2+4 a \sqrt {b} c d x-2 b^{3/2} c \sqrt {1-(c+d x)^2}+4 b^{3/2} c^2 \arcsin (c+d x)+4 b^{3/2} c d x \arcsin (c+d x)+4 a \sqrt {b} \cos (2 \arcsin (c+d x))+4 b^{3/2} \arcsin (c+d x) \cos (2 \arcsin (c+d x))-4 c \sqrt {2 \pi } (a+b \arcsin (c+d x))^{3/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+8 \sqrt {\pi } (a+b \arcsin (c+d x))^{3/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-4 c \sqrt {2 \pi } (a+b \arcsin (c+d x))^{3/2} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )-8 \sqrt {\pi } (a+b \arcsin (c+d x))^{3/2} \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )+b^{3/2} \sin (2 \arcsin (c+d x))}{3 b^{5/2} d^2 (a+b \arcsin (c+d x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(724\) vs. \(2(312)=624\).
Time = 1.16 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.89
method | result | size |
default | \(\frac {4 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) b c -4 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \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 ) b c +4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) a c -4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \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 ) a c +8 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +8 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a +8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a +4 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b c -4 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b c +4 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a c +\sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -4 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a}{3 d^{2} b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(725\) |
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Exception generated. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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