Integrand size = 14, antiderivative size = 204 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=-\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d} \]
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Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4887, 4715, 4767, 4809, 3387, 3386, 3432, 3385, 3433} \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4715
Rule 4767
Rule 4809
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}-\frac {(5 b) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d} \\ & = \frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {\left (15 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{8 d} \\ & = -\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{8 d} \\ & = -\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {\left (15 b^2 \cos \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 )}{8 d}-\frac {\left (15 b^2 \sin \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 )}{8 d} \\ & = -\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {\left (15 b^2 \cos \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 )}{4 d}-\frac {\left (15 b^2 \sin \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 )}{4 d} \\ & = -\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.35 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.05 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (-15 b (c+d x)+10 a \sqrt {1-c^2-2 c d x-d^2 x^2}+2 \left (4 a (c+d x)+5 b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)+4 b (c+d x) \arcsin (c+d x)^2\right )+2 a^2 \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+2 a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{16 d \sqrt {a+b \arcsin (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(164)=328\).
Time = 0.33 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {15 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b^{3}+15 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b^{3}+8 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}-40 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(441\) |
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Exception generated. \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\int \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.26 (sec) , antiderivative size = 1279, normalized size of antiderivative = 6.27 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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