Integrand size = 25, antiderivative size = 442 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {32 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d}+\frac {32 e^3 \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{15 b^{7/2} d} \] Output:
-2/5*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^(5/2)-4/5*e ^3*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+16/15*e^3*(d*x+c)^4/b^2/d/(a+ b*arcsin(d*x+c))^(3/2)-16/5*e^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arc sin(d*x+c))^(1/2)+128/15*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcs in(d*x+c))^(1/2)+32/15*e^3*2^(1/2)*Pi^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/ Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d-16/15*e^3*Pi^(1/2)*c os(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))/b^(7/2)/d -16/15*e^3*Pi^(1/2)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2)) *sin(2*a/b)/b^(7/2)/d+32/15*e^3*2^(1/2)*Pi^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/ 2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)/b^(7/2)/d
Result contains complex when optimal does not.
Time = 1.63 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.01 \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\frac {e^3 \left (-4 (a+b \arcsin (c+d x)) \left (e^{2 i \arcsin (c+d x)} (4 i a+b+4 i b \arcsin (c+d x))+4 \sqrt {2} b e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+e^{-2 i \arcsin (c+d x)} \left (-4 i a+b-4 i b \arcsin (c+d x)+4 \sqrt {2} b e^{\frac {2 i (a+b \arcsin (c+d x))}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )+4 (a+b \arcsin (c+d x)) \left (e^{-4 i \arcsin (c+d x)} (-8 i a+b-8 i b \arcsin (c+d x))+e^{4 i \arcsin (c+d x)} (8 i a+b+8 i b \arcsin (c+d x))+16 b e^{-\frac {4 i a}{b}} \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+16 b e^{\frac {4 i a}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )-6 b^2 \sin (2 \arcsin (c+d x))+3 b^2 \sin (4 \arcsin (c+d x))\right )}{60 b^3 d (a+b \arcsin (c+d x))^{5/2}} \] Input:
Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2),x]
Output:
(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((2*I)*ArcSin[c + d*x])*((4*I)*a + b + (4*I)*b*ArcSin[c + d*x]) + (4*Sqrt[2]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b )^(3/2)*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) + ((-4*I)*a + b - (4*I)*b*ArcSin[c + d*x] + 4*Sqrt[2]*b*E^(((2*I)*(a + b*Arc Sin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((2*I)* (a + b*ArcSin[c + d*x]))/b])/E^((2*I)*ArcSin[c + d*x])) + 4*(a + b*ArcSin[ c + d*x])*(((-8*I)*a + b - (8*I)*b*ArcSin[c + d*x])/E^((4*I)*ArcSin[c + d* x]) + E^((4*I)*ArcSin[c + d*x])*((8*I)*a + b + (8*I)*b*ArcSin[c + d*x]) + (16*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-4*I)*(a + b*A rcSin[c + d*x]))/b])/E^(((4*I)*a)/b) + 16*b*E^(((4*I)*a)/b)*((I*(a + b*Arc Sin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]) - 6 *b^2*Sin[2*ArcSin[c + d*x]] + 3*b^2*Sin[4*ArcSin[c + d*x]]))/(60*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))
Time = 2.28 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.20, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {5304, 27, 5144, 5222, 5142, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {8 \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \int \frac {c+d x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \int \frac {(c+d x)^3}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5142 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \int \left (\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (2 \sin \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {e^3 \left (\frac {6 \left (\frac {4 \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {8 \left (\frac {8 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^3 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^4}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2),x]
Output:
(e^3*((-2*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(5*b*(a + b*ArcSin[c + d*x])^ (5/2)) + (6*((-2*(c + d*x)^2)/(3*b*(a + b*ArcSin[c + d*x])^(3/2)) + (4*((- 2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) + (2*(S qrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqr t[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]] )/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b]))/b^2))/(3*b)))/(5*b) - (8*((-2*(c + d* x)^4)/(3*b*(a + b*ArcSin[c + d*x])^(3/2)) + (8*((-2*(c + d*x)^3*Sqrt[1 - ( c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) + (2*(-1/2*(Sqrt[b]*Sqrt[Pi/2 ]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b] ]) + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x] ])/(Sqrt[b]*Sqrt[Pi])])/2 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSi n[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/2 - (Sqrt[b]*Sqrt[Pi/2]*Fre snelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/2) )/b^2))/(3*b)))/(5*b)))/d
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[((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] - Simp [1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_)*((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. \(1246\) vs. \(2(368)=736\).
Time = 0.16 (sec) , antiderivative size = 1247, normalized size of antiderivative = 2.82
Input:
int((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/60/d*e^3/b^3*(-128*arcsin(d*x+c)^2*cos(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)*(a+b*arcsin(d*x+ c))^(1/2)*2^(1/2)*Pi^(1/2)*b^2+128*arcsin(d*x+c)^2*sin(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)*(a+ b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*b^2+64*arcsin(d*x+c)^2*cos(2*a/b)* FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b *arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b^2-64*arcsin(d*x+c)^2*sin(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+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*b^2-256*arcsin(d*x+c)*cos( 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)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*a*b+256*arcsin( d*x+c)*sin(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)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*a*b+ 128*arcsin(d*x+c)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b *arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*a *b-128*arcsin(d*x+c)*sin(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+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2 )*a*b-128*cos(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)*(a+b*arcsin(d*x+c))^(1/2)*2^(1/2)*Pi^(1/2)*a ^2+128*sin(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(...
Exception generated. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**3/(a+b*asin(d*x+c))**(7/2),x)
Output:
e**3*(Integral(c**3/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b* asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral (d**3*x**3/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral(3*c*d**2 *x**2/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x)) *asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b** 3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral(3*c**2*d*x/(a **3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x))
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((d*e*x + c*e)^3/(b*arcsin(d*x + c) + a)^(7/2), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^3/(b*arcsin(d*x + c) + a)^(7/2), x)
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:
int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(7/2),x)
Output:
int((c*e + d*e*x)^3/(a + b*asin(c + d*x))^(7/2), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {\left (d e x +c e \right )^{3}}{\left (\mathit {asin} \left (d x +c \right ) b +a \right )^{\frac {7}{2}}}d x \] Input:
int((d*e*x+c*e)^3/(a+b*asin(d*x+c))^(7/2),x)
Output:
int((d*e*x+c*e)^3/(a+b*asin(d*x+c))^(7/2),x)