Integrand size = 18, antiderivative size = 535 \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}-\frac {\sqrt {b} c \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {b} c^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d^3}+\frac {\sqrt {b} c^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d^3}-\frac {\sqrt {b} c \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 d^3} \]
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Time = 1.63 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4889, 4831, 6873, 6874, 3467, 3434, 3433, 3432, 3466, 3435, 3524, 3438} \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c^2 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^3}+\frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{12 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{4 d^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \cos (2 \arcsin (c+d x)) \sqrt {a+b \arcsin (c+d x)}}{2 d^3} \]
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Rule 3432
Rule 3433
Rule 3434
Rule 3435
Rule 3438
Rule 3466
Rule 3467
Rule 3524
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 )^2 \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 )^2 \, 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 )^2 \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^3} \\ & = \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 )^2 \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \left (c^2 x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right )+c x^2 \sin \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int x^2 \cos \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b d^3}+\frac {(2 c) \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^3}+\frac {\left (2 c^2\right ) \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^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}+\frac {\text {Subst}\left (\int \sin ^3\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 d^3}-\frac {c \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 )}{2 d^3}+\frac {c^2 \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^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}+\frac {\text {Subst}\left (\int \left (-\frac {1}{4} \sin \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )+\frac {3}{4} \sin \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{3 d^3}-\frac {\left (c^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 )}{d^3}-\frac {\left (c \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 )}{2 d^3}+\frac {\left (c^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 )}{d^3}-\frac {\left (c \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 )}{2 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} c^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 )}{d^3}+\frac {\sqrt {b} c^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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\text {Subst}\left (\int \sin \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{12 d^3}+\frac {\text {Subst}\left (\int \sin \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} c^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 )}{d^3}+\frac {\sqrt {b} c^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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^3}+\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{12 d^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{4 d^3}-\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{12 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d^3}+\frac {c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{2 d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} \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^3}-\frac {\sqrt {b} c^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 )}{d^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{12 d^3}+\frac {\sqrt {b} \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^3}+\frac {\sqrt {b} c^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 )}{d^3}-\frac {\sqrt {b} c \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 )}{4 d^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 d^3} \\ \end{align*}
Time = 3.25 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.86 \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {18 (c+d x) \sqrt {a+b \arcsin (c+d x)}+72 c^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}+36 c \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))-18 \sqrt {b} c \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-9 \sqrt {b} \left (1+4 c^2\right ) \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\sqrt {b} \sqrt {6 \pi } \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )+36 \sqrt {b} c^2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )-18 \sqrt {b} c \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )-\sqrt {b} \sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )-6 \sqrt {a+b \arcsin (c+d x)} \sin (3 \arcsin (c+d x))}{72 d^3} \]
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Time = 1.25 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {36 \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \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 ) b \,c^{2}+36 \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {1}{b}}\, \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^{2}-\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 -\sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \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 ) b +9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 -18 \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \sqrt {a +b \arcsin \left (d x +c \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 c +18 \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 c -72 \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}+36 \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 c -72 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,c^{2}+6 \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 ) b -18 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b +36 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a c +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a -18 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a}{72 d^{3} \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(777\) |
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Exception generated. \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\int x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx \]
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\[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\int { \sqrt {b \arcsin \left (d x + c\right ) + a} x^{2} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 2255, normalized size of antiderivative = 4.21 \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\int x^2\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]
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