Integrand size = 25, antiderivative size = 270 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} d}+\frac {e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} d}-\frac {e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2} d} \]
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Time = 0.31 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4889, 12, 4727, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} e^3 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\pi } e^3 \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} d}+\frac {\sqrt {\pi } e^3 \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} d}-\frac {\sqrt {\frac {\pi }{2}} e^3 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}} \]
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4727
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^3 x^3}{(a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^3 \text {Subst}\left (\int \frac {x^3}{(a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{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 )}{b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {e^3 \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 )}{b^2 d}+\frac {e^3 \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 )}{b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (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 )}{b^2 d}-\frac {\left (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 )}{b^2 d}+\frac {\left (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 )}{b^2 d}-\frac {\left (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 )}{b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (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 )}{b^2 d}-\frac {\left (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 )}{b^2 d}+\frac {\left (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 )}{b^2 d}-\frac {\left (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 )}{b^2 d} \\ & = -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {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 )}{b^{3/2} d}+\frac {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 )}{b^{3/2} d}+\frac {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 )}{b^{3/2} d}-\frac {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 )}{b^{3/2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.11 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {i e^3 e^{-\frac {4 i a}{b}} \left (\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {2} e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )-2 i e^{\frac {4 i a}{b}} \sin (2 \arcsin (c+d x))+i e^{\frac {4 i a}{b}} \sin (4 \arcsin (c+d x))\right )}{4 b d \sqrt {a+b \arcsin (c+d x)}} \]
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Time = 1.17 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {e^{3} \left (-2 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {-\frac {1}{b}}+2 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {-\frac {1}{b}}+4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+2 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right )-\sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right )\right )}{4 d b \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(329\) |
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Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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