Integrand size = 25, antiderivative size = 475 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d} \]
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Time = 1.00 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4889, 12, 4725, 4795, 4737, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {5 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {15 b e^3 \sqrt {1-(c+d x)^2} (c+d x) (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d} \]
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3432
Rule 3433
Rule 4725
Rule 4737
Rule 4795
Rule 4809
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^3 x^3 (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int x^3 (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}-\frac {\left (5 b e^3\right ) \text {Subst}\left (\int \frac {x^4 (a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = \frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int x^3 \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{64 d} \\ & = -\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int x \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{128 d}+\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{512 d} \\ & = -\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\sin ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d}+\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{512 d} \\ & = -\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{512 d}+\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d} \\ & = \frac {45 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4096 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{1024 d}+\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{512 d} \\ & = \frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{1024 d}-\frac {\left (15 b^2 e^3 \cos \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 )}{1024 d}+\frac {\left (15 b^2 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4096 d}-\frac {\left (15 b^2 e^3 \sin \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 )}{1024 d}+\frac {\left (15 b^2 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{4096 d} \\ & = \frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}-\frac {\left (15 b^2 e^3 \cos \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 )}{512 d}-\frac {\left (45 b^2 e^3 \cos \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 )}{1024 d}+\frac {\left (15 b^2 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2048 d}-\frac {\left (15 b^2 e^3 \sin \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 )}{512 d}-\frac {\left (45 b^2 e^3 \sin \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 )}{1024 d}+\frac {\left (15 b^2 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{2048 d} \\ & = \frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{1024 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{1024 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d}-\frac {\left (45 b^2 e^3 \cos \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 )}{512 d}-\frac {\left (45 b^2 e^3 \sin \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 )}{512 d} \\ & = \frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.54 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {i b^3 e^3 e^{-\frac {4 i a}{b}} \left (16 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-16 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )}{2048 d \sqrt {a+b \arcsin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(885\) vs. \(2(391)=782\).
Time = 1.23 (sec) , antiderivative size = 886, normalized size of antiderivative = 1.87
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Exception generated. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=e^{3} \left (\int a^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 a^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 a^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 3 b^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 6 a b c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 6 a b c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 3408, normalized size of antiderivative = 7.17 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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