Integrand size = 16, antiderivative size = 313 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\frac {b \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{6 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 c^3}+\frac {b^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{24 c^3} \] Output:
1/3*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c^3+1/6*b*x^2*(-c^2*x^2+1 )^(1/2)*(a+b*arcsin(c*x))^(1/2)/c+1/3*x^3*(a+b*arcsin(c*x))^(3/2)-3/16*b^( 3/2)*2^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x)) ^(1/2)/b^(1/2))/c^3+1/144*b^(3/2)*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelC(6^( 1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/c^3-3/16*b^(3/2)*2^(1/2)*Pi ^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b) /c^3+1/144*b^(3/2)*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin( c*x))^(1/2)/b^(1/2))*sin(3*a/b)/c^3
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\frac {b e^{-\frac {3 i a}{b}} \sqrt {a+b \arcsin (c x)} \left (27 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} \left (\sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{216 c^3 \sqrt {\frac {(a+b \arcsin (c x))^2}{b^2}}} \] Input:
Integrate[x^2*(a + b*ArcSin[c*x])^(3/2),x]
Output:
(b*Sqrt[a + b*ArcSin[c*x]]*(27*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]) )/b]*Gamma[5/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[(( -I)*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3 ]*(Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((-3*I)*(a + b*ArcSin[c*x])) /b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((3*I) *(a + b*ArcSin[c*x]))/b])))/(216*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcSin[c* x])^2/b^2])
Time = 2.70 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {5140, 5210, 5146, 4906, 2009, 5182, 5134, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 (a+b \arcsin (c x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 5140 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \int \frac {x^3 \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}}dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int \frac {x^2}{\sqrt {a+b \arcsin (c x)}}dx}{6 c}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {\int \left (\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \arcsin (c x)}}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {b \int \frac {1}{\sqrt {a+b \arcsin (c x)}}dx}{2 c}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 5134 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}+\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^{3/2}-\frac {1}{2} b c \left (\frac {\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {2 \left (\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c^2}-\frac {\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{3 c^2}\right )\) |
Input:
Int[x^2*(a + b*ArcSin[c*x])^(3/2),x]
Output:
(x^3*(a + b*ArcSin[c*x])^(3/2))/3 - (b*c*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*Sqrt [a + b*ArcSin[c*x]])/c^2 + (2*(-((Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x] ])/c^2) + (Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*Arc Sin[c*x]])/Sqrt[b]] + Sqrt[b]*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*A rcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c^2)))/(3*c^2) + ((Sqrt[b]*Sqrt[Pi/2]*C os[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/2 - (Sqrt[ b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/S qrt[b]])/2 + (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c* x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[ a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(6*c^4)))/2
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) Su bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(241)=482\).
Time = 0.15 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.75
| method | result | size |
| default | \(-\frac {27 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{2}-27 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{2}-\sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, b^{2}+\sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, b^{2}+36 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}-12 \arcsin \left (c x \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+72 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -54 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}-24 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b +6 \arcsin \left (c x \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b^{2}+36 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}-54 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -12 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2}+6 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a b}{144 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\) | \(547\) |
Input:
int(x^2*(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/144/c^3*(27*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a /b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2- 27*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelS (2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^2-(a+b*arcsin( c*x))^(1/2)*cos(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsi n(c*x))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*b^2+(a+b*arcsin(c*x))^(1/2) *sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/ 2)/b)*(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*b^2+36*arcsin(c*x)^2*sin(-(a+b*arcsin( c*x))/b+a/b)*b^2-12*arcsin(c*x)^2*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2+72 *arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b-54*arcsin(c*x)*cos(-(a+b*ar csin(c*x))/b+a/b)*b^2-24*arcsin(c*x)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b +6*arcsin(c*x)*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*b^2+36*sin(-(a+b*arcsin(c *x))/b+a/b)*a^2-54*cos(-(a+b*arcsin(c*x))/b+a/b)*a*b-12*sin(-3*(a+b*arcsin (c*x))/b+3*a/b)*a^2+6*cos(-3*(a+b*arcsin(c*x))/b+3*a/b)*a*b)/(a+b*arcsin(c *x))^(1/2)
Exception generated. \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(a+b*asin(c*x))**(3/2),x)
Output:
Integral(x**2*(a + b*asin(c*x))**(3/2), x)
\[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int { {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")
Output:
integrate((b*arcsin(c*x) + a)^(3/2)*x^2, x)
Result contains complex when optimal does not.
Time = 1.52 (sec) , antiderivative size = 2295, normalized size of antiderivative = 7.33 \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")
Output:
1/4*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b) /((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/8*I*sqrt(2)*sqrt(pi)*a* b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)* sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b ^2*sqrt(abs(b)))*c^3) + 1/4*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*sqr t(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sq rt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 1/8*I*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqr t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b) /((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c^3) - 1/2*sqrt(pi)*a^2*b^(3/2) *erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*a rcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/ abs(b))*c^3) - 1/12*I*sqrt(pi)*a*b^(5/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c* x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^ (3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c^3) - 1/8*sqrt(2)*sqrt(pi )*a^2*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt (2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/8*I*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I*sqrt(2)* sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) +...
Timed out. \[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2} \,d x \] Input:
int(x^2*(a + b*asin(c*x))^(3/2),x)
Output:
int(x^2*(a + b*asin(c*x))^(3/2), x)
\[ \int x^2 (a+b \arcsin (c x))^{3/2} \, dx=\left (\int \sqrt {\mathit {asin} \left (c x \right ) b +a}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) b +\left (\int \sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}d x \right ) a \] Input:
int(x^2*(a+b*asin(c*x))^(3/2),x)
Output:
int(sqrt(asin(c*x)*b + a)*asin(c*x)*x**2,x)*b + int(sqrt(asin(c*x)*b + a)* x**2,x)*a