Integrand size = 25, antiderivative size = 361 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 b^{3/2} e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 d} \]
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Time = 0.59 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {4889, 12, 4725, 4795, 4767, 4719, 3387, 3386, 3432, 3385, 3433, 4731, 4491} \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^2 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {\sqrt {\frac {\pi }{6}} b^{3/2} e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{24 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d} \]
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4719
Rule 4725
Rule 4731
Rule 4767
Rule 4795
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d} \\ & = \frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{12 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{12 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{6 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{12 d}-\frac {\left (b e^2\right ) \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 )}{6 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{48 d}-\frac {\left (b e^2\right ) \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 )}{48 d}-\frac {\left (b e^2 \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 )}{6 d}-\frac {\left (b e^2 \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 )}{6 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {\left (b e^2 \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 )}{48 d}-\frac {\left (b e^2 \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 d}+\frac {\left (b e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{48 d}-\frac {\left (b e^2 \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 )}{48 d}-\frac {\left (b e^2 \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 d}+\frac {\left (b e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{48 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{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 )}{3 d}-\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{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 )}{3 d}-\frac {\left (b e^2 \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 )}{24 d}+\frac {\left (b e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{24 d}-\frac {\left (b e^2 \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 )}{24 d}+\frac {\left (b e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{24 d} \\ & = \frac {b e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{3 d}+\frac {b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 \sqrt {\frac {\pi }{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 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{24 d}-\frac {3 b^{3/2} e^2 \sqrt {\frac {\pi }{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 d}+\frac {b^{3/2} e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.74 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\frac {b e^2 e^{-\frac {3 i a}{b}} \sqrt {a+b \arcsin (c+d x)} \left (27 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {3} \left (\sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {5}{2},\frac {3 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{216 d \sqrt {\frac {(a+b \arcsin (c+d x))^2}{b^2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(599\) vs. \(2(289)=578\).
Time = 1.21 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {e^{2} \left (-\sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{2}+\sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{2}+27 \sqrt {a +b \arcsin \left (d x +c \right )}\, \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}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-27 \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 ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}+36 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}-12 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+72 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b -54 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}-24 \arcsin \left (d x +c \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a b +6 \arcsin \left (d x +c \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+36 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2}-54 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b -12 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2}+6 \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a b \right )}{144 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(600\) |
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Exception generated. \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=e^{2} \left (\int a c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 a c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 b c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.56 (sec) , antiderivative size = 2199, normalized size of antiderivative = 6.09 \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c e+d e x)^2 (a+b \arcsin (c+d x))^{3/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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