Integrand size = 29, antiderivative size = 651 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {160 b^2 d^2 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {80 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11907 c^4}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1323 c^4}+\frac {50 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{27783 c^4}-\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \arcsin (c x)}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{189 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{21 \sqrt {1-c^2 x^2}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{441 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{81 \sqrt {1-c^2 x^2}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \]
5/63*d*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2+1/9*x^4*(-c^2*d*x^2+d) ^(5/2)*(a+b*arcsin(c*x))^2+160/3969*b^2*d^2*(-c^2*d*x^2+d)^(1/2)/c^4+80/11 907*b^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^4+4/1323*b^2*d^2*(-c^2*x^2 +1)^2*(-c^2*d*x^2+d)^(1/2)/c^4+50/27783*b^2*d^2*(-c^2*x^2+1)^3*(-c^2*d*x^2 +d)^(1/2)/c^4-2/729*b^2*d^2*(-c^2*x^2+1)^4*(-c^2*d*x^2+d)^(1/2)/c^4-2/63*d ^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4-1/63*d^2*x^2*(a+b*arcsin(c *x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+1/21*d^2*x^4*(a+b*arcsin(c*x))^2*(-c^2*d*x ^2+d)^(1/2)+4/63*a*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+4/6 3*b^2*d^2*x*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+2/189* b*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/21 *b*c*d^2*x^5*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+38/ 441*b*c^3*d^2*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2 )-2/81*b*c^5*d^2*x^9*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^( 1/2)
Time = 0.27 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.41 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {d^2 \sqrt {d-c^2 d x^2} \left (3969 a^2 \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right )+126 a b c x \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \sqrt {1-c^2 x^2} \left (-6140+899 c^2 x^2+1005 c^4 x^4-1147 c^6 x^6+343 c^8 x^8\right )+126 b \left (63 a \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right )+b c x \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )\right ) \arcsin (c x)+3969 b^2 \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right ) \arcsin (c x)^2\right )}{250047 c^4 \sqrt {1-c^2 x^2}} \]
-1/250047*(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(1 - c^2*x^2)^(7/2)*(2 + 7*c^ 2*x^2) + 126*a*b*c*x*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c ^8*x^8) + 2*b^2*Sqrt[1 - c^2*x^2]*(-6140 + 899*c^2*x^2 + 1005*c^4*x^4 - 11 47*c^6*x^6 + 343*c^8*x^8) + 126*b*(63*a*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2 ) + b*c*x*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c^8*x^8))*Ar cSin[c*x] + 3969*b^2*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2)*ArcSin[c*x]^2))/( c^4*Sqrt[1 - c^2*x^2])
Time = 3.65 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.23, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {5202, 5192, 27, 1578, 1195, 2009, 5202, 5192, 27, 354, 86, 2009, 5198, 5138, 243, 53, 2009, 5210, 5138, 243, 53, 2009, 5182, 2009}
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.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{9 \sqrt {1-c^2 x^2}}+\frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{315 \sqrt {1-c^2 x^2}}dx+\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{315} b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{630} b c \int \frac {x^4 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{630} b c \int \left (\frac {35 \left (1-c^2 x^2\right )^{7/2}}{c^4}-\frac {50 \left (1-c^2 x^2\right )^{5/2}}{c^4}+\frac {3 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {4 \sqrt {1-c^2 x^2}}{c^4}+\frac {8}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{7 \sqrt {1-c^2 x^2}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \frac {x^4 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \left (\frac {5 \left (1-c^2 x^2\right )^{5/2}}{c^4}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {\sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \arcsin (c x))dx}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {1-c^2 x^2}}dx\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \left (\frac {\left (1-c^2 x^2\right )^{3/2}}{c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {1}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \int x^2 (a+b \arcsin (c x))dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \arcsin (c x)) x^9-\frac {2}{7} c^2 (a+b \arcsin (c x)) x^7+\frac {1}{5} (a+b \arcsin (c x)) x^5-\frac {1}{630} b c \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{9 \sqrt {1-c^2 x^2}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \arcsin (c x)) x^7+\frac {1}{5} (a+b \arcsin (c x)) x^5-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}+\frac {2 \left (\frac {2 b \left (x \arcsin (c x) b+\frac {\sqrt {1-c^2 x^2} b}{c}+a x\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}\right )\right )\) |
(x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/9 - (2*b*c*d^2*Sqrt[d - c^2*d*x^2]*(-1/630*(b*c*((-16*Sqrt[1 - c^2*x^2])/c^6 - (8*(1 - c^2*x^2)^(3 /2))/(3*c^6) - (6*(1 - c^2*x^2)^(5/2))/(5*c^6) + (100*(1 - c^2*x^2)^(7/2)) /(7*c^6) - (70*(1 - c^2*x^2)^(9/2))/(9*c^6))) + (x^5*(a + b*ArcSin[c*x]))/ 5 - (2*c^2*x^7*(a + b*ArcSin[c*x]))/7 + (c^4*x^9*(a + b*ArcSin[c*x]))/9))/ (9*Sqrt[1 - c^2*x^2]) + (5*d*((x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x ])^2)/7 - (2*b*c*d*Sqrt[d - c^2*d*x^2]*(-1/70*(b*c*((-4*Sqrt[1 - c^2*x^2]) /c^6 - (2*(1 - c^2*x^2)^(3/2))/(3*c^6) + (16*(1 - c^2*x^2)^(5/2))/(5*c^6) - (10*(1 - c^2*x^2)^(7/2))/(7*c^6))) + (x^5*(a + b*ArcSin[c*x]))/5 - (c^2* x^7*(a + b*ArcSin[c*x]))/7))/(7*Sqrt[1 - c^2*x^2]) + (3*d*((x^4*Sqrt[d - c ^2*d*x^2]*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*Sqrt[d - c^2*d*x^2]*(-1/10*(b* c*((-2*Sqrt[1 - c^2*x^2])/c^6 + (4*(1 - c^2*x^2)^(3/2))/(3*c^6) - (2*(1 - c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSin[c*x]))/5))/(5*Sqrt[1 - c^2* x^2]) + (Sqrt[d - c^2*d*x^2]*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c* x])^2)/c^2 + (2*b*(-1/6*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x^2 )^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSin[c*x]))/3))/(3*c) + (2*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2) + (2*b*(a*x + (b*Sqrt[1 - c^2*x^2])/ c + b*x*ArcSin[c*x]))/c))/(3*c^2)))/(5*Sqrt[1 - c^2*x^2])))/7))/9
3.3.26.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ (a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c ^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 ] && IGtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*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] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -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]
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 2146, normalized size of antiderivative = 3.30
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2146\) |
| parts | \(\text {Expression too large to display}\) | \(2146\) |
a^2*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2))+ b^2*(1/373248*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8-256*I*(-c^ 2*x^2+1)^(1/2)*x^9*c^9+688*c^6*x^6+576*I*(-c^2*x^2+1)^(1/2)*x^7*c^7-280*c^ 4*x^4-432*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+41*c^2*x^2+120*I*(-c^2*x^2+1)^(1/2) *x^3*c^3-9*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(18*I*arcsin(c*x)+81*arcsin(c*x)^2- 2)*d^2/c^4/(c^2*x^2-1)-3/175616*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6 *x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)* x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)* x*c+1)*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^4/(c^2*x^2-1)+1/1728*(- d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3 *I*(-c^2*x^2+1)^(1/2)*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d^2/c^4/( c^2*x^2-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c- 1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))*d^2/c^4/(c^2*x^2-1)-3/256*(-d*(c^2*x^ 2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcs in(c*x))*d^2/c^4/(c^2*x^2-1)+1/1728*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(- c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-6*I*a rcsin(c*x)+9*arcsin(c*x)^2-2)*d^2/c^4/(c^2*x^2-1)-3/175616*(-d*(c^2*x^2-1) )^(1/2)*(64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/ 2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c ^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*(-14*I*arcsin(c*x)+49*arcsin(c*x)^2-2...
Time = 0.29 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.75 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {126 \, {\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x + {\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (343 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \, {\left (51597 \, a^{2} - 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} - 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \, {\left (15876 \, a^{2} - 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} - 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \, {\left (3969 \, a^{2} - 6140 \, b^{2}\right )} d^{2} + 3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 7938 \, {\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
1/250047*(126*(49*a*b*c^9*d^2*x^9 - 171*a*b*c^7*d^2*x^7 + 189*a*b*c^5*d^2* x^5 - 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2*x + (49*b^2*c^9*d^2*x^9 - 171*b^2 *c^7*d^2*x^7 + 189*b^2*c^5*d^2*x^5 - 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x) *arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (343*(81*a^2 - 2*b ^2)*c^10*d^2*x^10 - 2*(51597*a^2 - 1490*b^2)*c^8*d^2*x^8 + 2*(67473*a^2 - 2152*b^2)*c^6*d^2*x^6 - 4*(15876*a^2 - 53*b^2)*c^4*d^2*x^4 - (3969*a^2 - 1 4078*b^2)*c^2*d^2*x^2 + 2*(3969*a^2 - 6140*b^2)*d^2 + 3969*(7*b^2*c^10*d^2 *x^10 - 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2*x^6 - 16*b^2*c^4*d^2*x^4 - b^2 *c^2*d^2*x^2 + 2*b^2*d^2)*arcsin(c*x)^2 + 7938*(7*a*b*c^10*d^2*x^10 - 26*a *b*c^8*d^2*x^8 + 34*a*b*c^6*d^2*x^6 - 16*a*b*c^4*d^2*x^4 - a*b*c^2*d^2*x^2 + 2*a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.62 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a^{2} - \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {-c^{2} x^{2} + 1} c^{6} d^{\frac {5}{2}} x^{8} - 1147 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {5}{2}} x^{6} + 1005 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} + 899 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} - \frac {6140 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {5}{2}}}{c^{2}}}{c^{2}} + \frac {63 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} a b}{3969 \, c^{3}} \]
-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^ 4*d))*b^2*arcsin(c*x)^2 - 2/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*( -c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*b*arcsin(c*x) - 1/63*(7*(-c^2*d*x^2 + d)^ (7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a^2 - 2/250047*b^2*( (343*sqrt(-c^2*x^2 + 1)*c^6*d^(5/2)*x^8 - 1147*sqrt(-c^2*x^2 + 1)*c^4*d^(5 /2)*x^6 + 1005*sqrt(-c^2*x^2 + 1)*c^2*d^(5/2)*x^4 + 899*sqrt(-c^2*x^2 + 1) *d^(5/2)*x^2 - 6140*sqrt(-c^2*x^2 + 1)*d^(5/2)/c^2)/c^2 + 63*(49*c^8*d^(5/ 2)*x^9 - 171*c^6*d^(5/2)*x^7 + 189*c^4*d^(5/2)*x^5 - 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*arcsin(c*x)/c^3) - 2/3969*(49*c^8*d^(5/2)*x^9 - 171*c^6*d^( 5/2)*x^7 + 189*c^4*d^(5/2)*x^5 - 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*a*b/c ^3
Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]