Integrand size = 16, antiderivative size = 269 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=-\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d^2}+\frac {\sqrt {b} c \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^2}-\frac {\sqrt {b} c \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d^2}+\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d^2} \]
[Out]
Time = 0.55 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4889, 4831, 6873, 6874, 3467, 3434, 3433, 3432, 3466, 3435} \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^2}+\frac {\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d^2}+\frac {\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d^2}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^2}-\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\cos (2 \arcsin (c+d x)) \sqrt {a+b \arcsin (c+d x)}}{4 d^2} \]
[In]
[Out]
Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3466
Rule 3467
Rule 4831
Rule 4889
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \sqrt {a+b x} \cos (x) \left (-\frac {c}{d}+\frac {\sin (x)}{d}\right ) \, dx,x,\arcsin (c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a-x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sin \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \left (c x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^2 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^2} \\ & = -\frac {\text {Subst}\left (\int x^2 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {\text {Subst}\left (\int \cos \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^2}-\frac {c \text {Subst}\left (\int \sin \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{d^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {\left (c \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 )}{d^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^2}-\frac {\left (c \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 )}{d^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {\sqrt {b} \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 )}{8 d^2}+\frac {\sqrt {b} c \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 )}{d^2}-\frac {\sqrt {b} c \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 )}{d^2}+\frac {\sqrt {b} \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 )}{8 d^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.92 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {-2 \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))+\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {4 b c e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{\sqrt {a+b \arcsin (c+d x)}}+\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d^2} \]
[In]
[Out]
Time = 1.11 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) b c +4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b c -\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -8 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b c +2 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -8 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a c +2 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a}{8 d^{2} \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(390\) |
[In]
[Out]
Exception generated. \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\int x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
\[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\int { \sqrt {b \arcsin \left (d x + c\right ) + a} x \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 1079, normalized size of antiderivative = 4.01 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\int x\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]
[In]
[Out]