Integrand size = 23, antiderivative size = 301 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {105 b^{7/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^{7/2} e \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{512 d} \]
35/64*b^2*e*(a+b*arcsin(d*x+c))^(3/2)/d-35/32*b^2*e*(d*x+c)^2*(a+b*arcsin( d*x+c))^(3/2)/d-1/4*e*(a+b*arcsin(d*x+c))^(7/2)/d+1/2*e*(d*x+c)^2*(a+b*arc sin(d*x+c))^(7/2)/d+105/512*b^(7/2)*e*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d* x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d-105/512*b^(7/2)*e*FresnelC(2*(a+b *arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+7/8*b*e*(d*x +c)*(a+b*arcsin(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2)/d-105/128*b^3*e*(d*x+c)* (1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.46 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {b^4 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {9}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {9}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )}{64 \sqrt {2} d \sqrt {a+b \arcsin (c+d x)}} \]
-1/64*(b^4*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x] ))/b]*Gamma[9/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(Sqrt[2]*d*E^(((2*I) *a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
Time = 2.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.96, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {5304, 27, 5140, 5210, 5140, 5152, 5210, 5146, 25, 4906, 27, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 5152}
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.
| \(\displaystyle \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \arcsin (c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \arcsin (c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5140 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \int (c+d x) (a+b \arcsin (c+d x))^{3/2}d(c+d x)+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^{5/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5140 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \int \frac {(c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^{5/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \int \frac {(c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{4} b \int \frac {c+d x}{\sqrt {a+b \arcsin (c+d x)}}d(c+d x)+\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5146 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{4} \int -\frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{8} \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{8} \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\cos \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (2 \cos \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{8} \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-2 \sin \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )+\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{2} \int \frac {\sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)+\frac {1}{8} \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )-\frac {1}{2} \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}-\frac {7}{4} b \left (\frac {5}{4} b \left (\frac {1}{2} (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}-\frac {3}{4} b \left (\frac {1}{8} \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )+\frac {(a+b \arcsin (c+d x))^{3/2}}{3 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )+\frac {(a+b \arcsin (c+d x))^{7/2}}{7 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )\right )}{d}\) |
(e*(((c + d*x)^2*(a + b*ArcSin[c + d*x])^(7/2))/2 - (7*b*(-1/2*((c + d*x)* Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2)) + (a + b*ArcSin[c + d *x])^(7/2)/(7*b) + (5*b*(((c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/2 - ( 3*b*(-1/2*((c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]]) + (a + b*ArcSin[c + d*x])^(3/2)/(3*b) + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*Fresn elS[(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]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b] )/8))/4))/4))/4))/d
3.3.56.3.1 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 + 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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -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]
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. \(653\) vs. \(2(249)=498\).
Time = 0.92 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.17
| method | result | size |
| default | \(\frac {e b \left (128 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+384 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+224 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+384 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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 ) a^{2} b -280 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \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^{3}+448 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+128 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-280 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+224 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -210 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}-105 \pi \,b^{3} \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 )-105 \pi \,b^{3} \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 )\right ) \sqrt {-\frac {1}{b}}}{512 d \sqrt {\pi }}\) | \(654\) |
1/512*e/d*b*(128*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d* x+c)^3*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+384*(a+b*arcsin(d*x+c))^(1/ 2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/ b)*a*b^2+224*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c) ^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+384*(a+b*arcsin(d*x+c))^(1/2)*P i^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2 *b-280*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)*cos(- 2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+448*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)* (-1/b)^(1/2)*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+128*( a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-2*(a+b*arcsin(d*x+c))/ b+2*a/b)*a^3-280*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-2*(a +b*arcsin(d*x+c))/b+2*a/b)*a*b^2+224*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(- 1/b)^(1/2)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-210*(a+b*arcsin(d*x+c ))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3-105 *Pi*b^3*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)-105*Pi*b^3*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))*(-1/b)^(1/2)/Pi^(1/2)
Exception generated. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 2561, normalized size of antiderivative = 8.51 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Too large to display} \]
-1/1024*(128*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)^3*e^(2*I*ar csin(d*x + c)) + 128*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)^3*e ^(-2*I*arcsin(d*x + c)) + 384*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin(d *x + c)^2*e^(2*I*arcsin(d*x + c)) + 224*I*sqrt(b*arcsin(d*x + c) + a)*b^3* e*arcsin(d*x + c)^2*e^(2*I*arcsin(d*x + c)) + 384*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin(d*x + c)^2*e^(-2*I*arcsin(d*x + c)) - 224*I*sqrt(b*arcs in(d*x + c) + a)*b^3*e*arcsin(d*x + c)^2*e^(-2*I*arcsin(d*x + c)) + 768*I* sqrt(pi)*a^4*b*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsi n(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b^(3/2) + I*b^(5/2)/abs(b)) + 192*sqrt(pi)*a^3*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt( b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b^(3/2) + I*b^(5/2)/ab s(b)) - 768*I*sqrt(pi)*a^4*b*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b^(3/2) - I*b^ (5/2)/abs(b)) + 192*sqrt(pi)*a^3*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sq rt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b^(3/2 ) - I*b^(5/2)/abs(b)) - 256*I*sqrt(pi)*a^4*sqrt(b)*e*erf(-sqrt(b*arcsin(d* x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I *a/b)/(b + I*b^2/abs(b)) + 832*sqrt(pi)*a^3*b^(3/2)*e*erf(-sqrt(b*arcsin(d *x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2* I*a/b)/(b + I*b^2/abs(b)) - 288*I*sqrt(pi)*a^2*b^(5/2)*e*erf(-sqrt(b*ar...
Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \]