Integrand size = 25, antiderivative size = 344 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {16 e^3 (c+d x)^4}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {4 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {4 e^3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d}-\frac {4 e^3 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{3 b^{5/2} d} \]
-4/3*e^3*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2)) *Pi^(1/2)/b^(5/2)/d+4/3*e^3*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/P i^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(5/2)/d+4/3*e^3*cos(4*a/b)*FresnelS(2*2^(1/ 2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/d- 4/3*e^3*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin (4*a/b)*2^(1/2)*Pi^(1/2)/b^(5/2)/d-2/3*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b /d/(a+b*arcsin(d*x+c))^(3/2)-4*e^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^(1/ 2)+16/3*e^3*(d*x+c)^4/b^2/d/(a+b*arcsin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 2.22 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.02 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {e^3 \left (-4 (a+b \arcsin (c+d x)) \left (e^{-2 i \arcsin (c+d x)}+e^{2 i \arcsin (c+d x)}-\sqrt {2} e^{-\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )+4 (a+b \arcsin (c+d x)) \left (e^{-4 i \arcsin (c+d x)}+e^{4 i \arcsin (c+d x)}-2 e^{-\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )-2 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )-2 b \sin (2 \arcsin (c+d x))+b \sin (4 \arcsin (c+d x))\right )}{12 b^2 d (a+b \arcsin (c+d x))^{3/2}} \]
(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((-2*I)*ArcSin[c + d*x]) + E^((2*I)*Ar cSin[c + d*x]) - (Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2 , ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((2*I) *a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin [c + d*x]))/b]) + 4*(a + b*ArcSin[c + d*x])*(E^((-4*I)*ArcSin[c + d*x]) + E^((4*I)*ArcSin[c + d*x]) - (2*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamm a[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((4*I)*a)/b) - 2*E^(((4*I)* a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[ c + d*x]))/b]) - 2*b*Sin[2*ArcSin[c + d*x]] + b*Sin[4*ArcSin[c + d*x]]))/( 12*b^2*d*(a + b*ArcSin[c + d*x])^(3/2))
Time = 1.82 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {5304, 27, 5144, 5222, 5146, 25, 4906, 27, 2009, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{b}-\frac {8 \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {4 \int \frac {c+d x}{\sqrt {a+b \arcsin (c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \int \frac {(c+d x)^3}{\sqrt {a+b \arcsin (c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {e^3 \left (-\frac {8 \left (\frac {8 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}+\frac {2 \left (\frac {4 \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (-\frac {8 \left (-\frac {8 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}+\frac {2 \left (-\frac {4 \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (-\frac {4 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (-\frac {8 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{4 \sqrt {a+b \arcsin (c+d x)}}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (-\frac {8 \int \left (\frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{4 \sqrt {a+b \arcsin (c+d x)}}-\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (-\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^2}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{b}-\frac {8 \left (\frac {8 \left (-\frac {1}{4} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x)^4}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{d}\) |
(e^3*((-2*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(3*b*(a + b*ArcSin[c + d*x])^ (3/2)) + (2*((-2*(c + d*x)^2)/(b*Sqrt[a + b*ArcSin[c + d*x]]) + (2*(Sqrt[b ]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]* Sqrt[Pi])] - Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sq rt[b]*Sqrt[Pi])]*Sin[(2*a)/b]))/b^2))/b - (8*((-2*(c + d*x)^4)/(b*Sqrt[a + b*ArcSin[c + d*x]]) + (8*(-1/8*(Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[ (2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]) + (Sqrt[b]*Sqrt[Pi]*C os[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/ 4 - (Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sq rt[Pi])]*Sin[(2*a)/b])/4 + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt [a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/8))/b^2))/(3*b)))/d
3.3.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Sim p[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt [1 - c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcSi n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ m, 0] && LtQ[n, -2]
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_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^ 2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Simp[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b* ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* d + e, 0] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(732\) vs. \(2(284)=568\).
Time = 1.30 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(\frac {e^{3} \left (-16 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \arcsin \left (d x +c \right ) b -16 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \arcsin \left (d x +c \right ) b +16 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 +16 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 -16 \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, a -16 \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, a +16 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 ) a +16 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 ) a -8 \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 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) \arcsin \left (d x +c \right ) b +2 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -8 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a -\sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b +8 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \right )}{12 d \,b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(733\) |
1/12*e^3/d/b^2*(-16*cos(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a +b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c ))^(1/2)*arcsin(d*x+c)*b-16*sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^ (1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcs in(d*x+c))^(1/2)*arcsin(d*x+c)*b+16*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a +b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2 )*(a+b*arcsin(d*x+c))^(1/2)/b)*b+16*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a +b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2 )*(a+b*arcsin(d*x+c))^(1/2)/b)*b-16*cos(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2) /(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*( a+b*arcsin(d*x+c))^(1/2)*a-16*sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b )^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*ar csin(d*x+c))^(1/2)*a+16*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*co s(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2 )/b)*a+16*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*Fresn elC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a-8*arcsi n(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b+8*cos(-4*(a+b*arcsin(d*x+c) )/b+4*a/b)*arcsin(d*x+c)*b+2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b-8*cos(- 2*(a+b*arcsin(d*x+c))/b+2*a/b)*a-sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b+8*c os(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a)/(a+b*arcsin(d*x+c))^(3/2)
Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asi n(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)** 2), x) + Integral(d**3*x**3/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2), x) + Integral(3*c*d**2*x**2/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*asin(c + d *x))*asin(c + d*x)**2), x) + Integral(3*c**2*d*x/(a**2*sqrt(a + b*asin(c + d*x)) + 2*a*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + b**2*sqrt(a + b*a sin(c + d*x))*asin(c + d*x)**2), x))
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]