Integrand size = 27, antiderivative size = 391 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {10516 b^2 d^3 x}{99225 c^2}-\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c^3}+\frac {32 b d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{945 c}+\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{315 c^3}+\frac {4 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{525 c^3}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^3}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \arcsin (c x))^2+\frac {8}{105} d^3 x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 \]
-10516/99225*b^2*d^3*x/c^2-5258/297675*b^2*d^3*x^3+4198/165375*b^2*c^2*d^3 *x^5-374/27783*b^2*c^4*d^3*x^7+2/729*b^2*c^6*d^3*x^9+16/315*b*d^3*(-c^2*x^ 2+1)^(3/2)*(a+b*arcsin(c*x))/c^3+4/525*b*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsi n(c*x))/c^3+2/441*b*d^3*(-c^2*x^2+1)^(7/2)*(a+b*arcsin(c*x))/c^3-2/81*b*d^ 3*(-c^2*x^2+1)^(9/2)*(a+b*arcsin(c*x))/c^3+16/315*d^3*x^3*(a+b*arcsin(c*x) )^2+8/105*d^3*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+2/21*d^3*x^3*(-c^2*x^2+ 1)^2*(a+b*arcsin(c*x))^2+1/9*d^3*x^3*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2+64 /945*b*d^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+32/945*b*d^3*x^2*(a+b* arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.71 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {d^3 \left (99225 a^2 c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+630 a b \sqrt {1-c^2 x^2} \left (-5258-2629 c^2 x^2+6297 c^4 x^4-4675 c^6 x^6+1225 c^8 x^8\right )+b^2 \left (3312540 c x+552090 c^3 x^3-793422 c^5 x^5+420750 c^7 x^7-85750 c^9 x^9\right )+630 b \left (315 a c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-5258-2629 c^2 x^2+6297 c^4 x^4-4675 c^6 x^6+1225 c^8 x^8\right )\right ) \arcsin (c x)+99225 b^2 c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right ) \arcsin (c x)^2\right )}{31255875 c^3} \]
-1/31255875*(d^3*(99225*a^2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35 *c^6*x^6) + 630*a*b*Sqrt[1 - c^2*x^2]*(-5258 - 2629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8) + b^2*(3312540*c*x + 552090*c^3*x^3 - 7934 22*c^5*x^5 + 420750*c^7*x^7 - 85750*c^9*x^9) + 630*b*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-5258 - 2 629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSin[c*x] + 9 9225*b^2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6)*ArcSin[c* x]^2))/c^3
Time = 2.50 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.37, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {5202, 27, 5194, 27, 290, 2009, 5202, 5194, 27, 290, 2009, 5202, 5138, 5194, 27, 2009, 5210, 15, 5182, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2}{9} b c d^3 \int x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {2}{3} d \int d^2 x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^3 \int x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^3 \left (-b c \int -\frac {\left (1-c^2 x^2\right )^3 \left (7 c^2 x^2+2\right )}{63 c^4}dx+\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^3 \left (\frac {b \int \left (1-c^2 x^2\right )^3 \left (7 c^2 x^2+2\right )dx}{63 c^3}+\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^3 \left (\frac {b \int \left (-7 c^8 x^8+19 c^6 x^6-15 c^4 x^4+c^2 x^2+2\right )dx}{63 c^3}+\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \int x^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{7} b c \int x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{7} b c \left (-b c \int -\frac {\left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )}{35 c^4}dx+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{7} b c \left (\frac {b \int \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{7} b c \left (\frac {b \int \left (5 c^6 x^6-8 c^4 x^4+c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \int x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (-\frac {2}{5} b c \int x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{5} \int x^2 (a+b \arcsin (c x))^2dx+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{5} b c \int x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{5} b c \left (-b c \int -\frac {-3 c^4 x^4+c^2 x^2+2}{15 c^4}dx+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{5} b c \left (\frac {b \int \left (-3 c^4 x^4+c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}\right )\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2}{3} d^3 \left (\frac {4}{7} \left (\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )+\frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2+\frac {2}{3} d^3 \left (\frac {1}{7} x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{7} \left (\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{5} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))^2-\frac {2}{3} b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^4}+\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )-\frac {2}{7} b c \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^4}+\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\right )-\frac {2}{9} b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {b \left (-\frac {7}{9} c^8 x^9+\frac {19 c^6 x^7}{7}-3 c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{63 c^3}\right )\) |
(d^3*x^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/9 - (2*b*c*d^3*((b*(2*x + (c^2*x^3)/3 - 3*c^4*x^5 + (19*c^6*x^7)/7 - (7*c^8*x^9)/9))/(63*c^3) - ((1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4) + ((1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^4)))/9 + (2*d^3*((x^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c *x])^2)/7 - (2*b*c*((b*(2*x + (c^2*x^3)/3 - (8*c^4*x^5)/5 + (5*c^6*x^7)/7) )/(35*c^3) - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^4) + ((1 - c^2 *x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4)))/7 + (4*((x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*((b*(2*x + (c^2*x^3)/3 - (3*c^4*x^5)/5))/(15 *c^3) - ((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^4) + ((1 - c^2*x^2) ^(5/2)*(a + b*ArcSin[c*x]))/(5*c^4)))/5 + (2*((x^3*(a + b*ArcSin[c*x])^2)/ 3 - (2*b*c*((b*x^3)/(9*c) - (x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3 *c^2) + (2*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/(3*c^2 )))/3))/5))/7))/3
3.2.76.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
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_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
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]
Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.34
| method | result | size |
| parts | \(-d^{3} a^{2} \left (\frac {1}{9} c^{6} x^{9}-\frac {3}{7} c^{4} x^{7}+\frac {3}{5} c^{2} x^{5}-\frac {1}{3} x^{3}\right )-\frac {d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )}{c^{3}}-\frac {2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(524\) |
| derivativedivides | \(\frac {-d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(525\) |
| default | \(\frac {-d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(525\) |
-d^3*a^2*(1/9*c^6*x^9-3/7*c^4*x^7+3/5*c^2*x^5-1/3*x^3)-d^3*b^2/c^3*(1/35*a rcsin(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x+2/441*arcsin(c*x)*(c ^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/15435*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35 )*c*x-4/525*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)+4/7875*(3*c^4*x^4 -10*c^2*x^2+15)*c*x+16/945*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/2 835*(c^2*x^2-3)*c*x+32/315*c*x-32/315*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+1/315 *arcsin(c*x)^2*(35*c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x+2/ 81*arcsin(c*x)*(c^2*x^2-1)^4*(-c^2*x^2+1)^(1/2)-2/25515*(35*c^8*x^8-180*c^ 6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x)-2*d^3*a*b/c^3*(1/9*arcsin(c*x)*c^9 *x^9-3/7*arcsin(c*x)*c^7*x^7+3/5*arcsin(c*x)*c^5*x^5-1/3*c^3*x^3*arcsin(c* x)+1/81*c^8*x^8*(-c^2*x^2+1)^(1/2)-187/3969*c^6*x^6*(-c^2*x^2+1)^(1/2)+209 9/33075*c^4*x^4*(-c^2*x^2+1)^(1/2)-2629/99225*c^2*x^2*(-c^2*x^2+1)^(1/2)-5 258/99225*(-c^2*x^2+1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.95 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {42875 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{3} x^{9} - 1125 \, {\left (11907 \, a^{2} - 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} - 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} - 105 \, {\left (99225 \, a^{2} - 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} + 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} - 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} - 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right )^{2} + 198450 \, {\left (35 \, a b c^{9} d^{3} x^{9} - 135 \, a b c^{7} d^{3} x^{7} + 189 \, a b c^{5} d^{3} x^{5} - 105 \, a b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + 630 \, {\left (1225 \, a b c^{8} d^{3} x^{8} - 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} - 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3} + {\left (1225 \, b^{2} c^{8} d^{3} x^{8} - 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} - 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{31255875 \, c^{3}} \]
-1/31255875*(42875*(81*a^2 - 2*b^2)*c^9*d^3*x^9 - 1125*(11907*a^2 - 374*b^ 2)*c^7*d^3*x^7 + 189*(99225*a^2 - 4198*b^2)*c^5*d^3*x^5 - 105*(99225*a^2 - 5258*b^2)*c^3*d^3*x^3 + 3312540*b^2*c*d^3*x + 99225*(35*b^2*c^9*d^3*x^9 - 135*b^2*c^7*d^3*x^7 + 189*b^2*c^5*d^3*x^5 - 105*b^2*c^3*d^3*x^3)*arcsin(c *x)^2 + 198450*(35*a*b*c^9*d^3*x^9 - 135*a*b*c^7*d^3*x^7 + 189*a*b*c^5*d^3 *x^5 - 105*a*b*c^3*d^3*x^3)*arcsin(c*x) + 630*(1225*a*b*c^8*d^3*x^8 - 4675 *a*b*c^6*d^3*x^6 + 6297*a*b*c^4*d^3*x^4 - 2629*a*b*c^2*d^3*x^2 - 5258*a*b* d^3 + (1225*b^2*c^8*d^3*x^8 - 4675*b^2*c^6*d^3*x^6 + 6297*b^2*c^4*d^3*x^4 - 2629*b^2*c^2*d^3*x^2 - 5258*b^2*d^3)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^ 3
Time = 1.63 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.60 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{6} d^{3} x^{9}}{9} + \frac {3 a^{2} c^{4} d^{3} x^{7}}{7} - \frac {3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac {a^{2} d^{3} x^{3}}{3} - \frac {2 a b c^{6} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{9} - \frac {2 a b c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} + \frac {6 a b c^{4} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {374 a b c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} - \frac {6 a b c^{2} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {4198 a b c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075} + \frac {2 a b d^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {5258 a b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c} + \frac {10516 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {b^{2} c^{6} d^{3} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac {2 b^{2} c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} - \frac {374 b^{2} c^{4} d^{3} x^{7}}{27783} + \frac {374 b^{2} c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3969} - \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} + \frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac {4198 b^{2} c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {5258 b^{2} d^{3} x^{3}}{297675} + \frac {5258 b^{2} d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c} - \frac {10516 b^{2} d^{3} x}{99225 c^{2}} + \frac {10516 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**6*d**3*x**9/9 + 3*a**2*c**4*d**3*x**7/7 - 3*a**2*c**2* d**3*x**5/5 + a**2*d**3*x**3/3 - 2*a*b*c**6*d**3*x**9*asin(c*x)/9 - 2*a*b* c**5*d**3*x**8*sqrt(-c**2*x**2 + 1)/81 + 6*a*b*c**4*d**3*x**7*asin(c*x)/7 + 374*a*b*c**3*d**3*x**6*sqrt(-c**2*x**2 + 1)/3969 - 6*a*b*c**2*d**3*x**5* asin(c*x)/5 - 4198*a*b*c*d**3*x**4*sqrt(-c**2*x**2 + 1)/33075 + 2*a*b*d**3 *x**3*asin(c*x)/3 + 5258*a*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(99225*c) + 10 516*a*b*d**3*sqrt(-c**2*x**2 + 1)/(99225*c**3) - b**2*c**6*d**3*x**9*asin( c*x)**2/9 + 2*b**2*c**6*d**3*x**9/729 - 2*b**2*c**5*d**3*x**8*sqrt(-c**2*x **2 + 1)*asin(c*x)/81 + 3*b**2*c**4*d**3*x**7*asin(c*x)**2/7 - 374*b**2*c* *4*d**3*x**7/27783 + 374*b**2*c**3*d**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x )/3969 - 3*b**2*c**2*d**3*x**5*asin(c*x)**2/5 + 4198*b**2*c**2*d**3*x**5/1 65375 - 4198*b**2*c*d**3*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/33075 + b**2* d**3*x**3*asin(c*x)**2/3 - 5258*b**2*d**3*x**3/297675 + 5258*b**2*d**3*x** 2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(99225*c) - 10516*b**2*d**3*x/(99225*c**2 ) + 10516*b**2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(99225*c**3), Ne(c, 0)) , (a**2*d**3*x**3/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 946 vs. \(2 (346) = 692\).
Time = 0.32 (sec) , antiderivative size = 946, normalized size of antiderivative = 2.42 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{9} \, b^{2} c^{6} d^{3} x^{9} \arcsin \left (c x\right )^{2} - \frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{4} d^{3} x^{7} \arcsin \left (c x\right )^{2} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, b^{2} c^{2} d^{3} x^{5} \arcsin \left (c x\right )^{2} - \frac {2}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} a b c^{6} d^{3} - \frac {2}{893025} \, {\left (315 \, {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c \arcsin \left (c x\right ) - \frac {1225 \, c^{8} x^{9} + 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} + 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{6} d^{3} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {6}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{3} + \frac {2}{8575} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{3} + \frac {1}{3} \, b^{2} d^{3} x^{3} \arcsin \left (c x\right )^{2} - \frac {2}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d^{3} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{3} + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{3} \]
-1/9*b^2*c^6*d^3*x^9*arcsin(c*x)^2 - 1/9*a^2*c^6*d^3*x^9 + 3/7*b^2*c^4*d^3 *x^7*arcsin(c*x)^2 + 3/7*a^2*c^4*d^3*x^7 - 3/5*b^2*c^2*d^3*x^5*arcsin(c*x) ^2 - 2/2835*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqr t(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c)*a*b*c^6*d^3 - 2/893025*(31 5*(35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt (-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c*arcsin(c*x) - (1225*c^8*x^9 + 1800*c^6*x^7 + 3024*c^4*x^5 + 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^6*d^3 - 3/5*a^2*c^2*d^3*x^5 + 6/245*(35 *x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^ 4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c ^4*d^3 + 2/8575*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)* x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcs in(c*x) - (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^4*d ^3 + 1/3*b^2*d^3*x^3*arcsin(c*x)^2 - 2/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c ^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/ c^6)*c)*a*b*c^2*d^3 - 2/375*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^ 2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arcsin(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d^3 + 1/3*a^2*d^3*x^3 + 2/9*(3*x^3*arcs in(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*...
Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (346) = 692\).
Time = 0.34 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.83 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{9 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x}{729 \, c^{2}} + \frac {1}{3} \, a^{2} d^{3} x^{3} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{3} x \arcsin \left (c x\right )}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{105 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{81 \, c^{3}} - \frac {622 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x}{250047 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{3} x \arcsin \left (c x\right )}{105 \, c^{2}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{81 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{441 \, c^{3}} + \frac {15224 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x}{10418625 \, c^{2}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )} a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {16 \, b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{441 \, c^{3}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{525 \, c^{3}} + \frac {115504 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {32 \, a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{525 \, c^{3}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} \arcsin \left (c x\right )}{945 \, c^{3}} - \frac {3406208 \, b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3}}{945 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{315 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{315 \, c^{3}} \]
-1/9*a^2*c^6*d^3*x^9 + 3/7*a^2*c^4*d^3*x^7 - 3/5*a^2*c^2*d^3*x^5 - 1/9*(c^ 2*x^2 - 1)^4*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/9*(c^2*x^2 - 1)^4*a*b*d^3*x*a rcsin(c*x)/c^2 - 1/63*(c^2*x^2 - 1)^3*b^2*d^3*x*arcsin(c*x)^2/c^2 + 2/729* (c^2*x^2 - 1)^4*b^2*d^3*x/c^2 + 1/3*a^2*d^3*x^3 - 2/63*(c^2*x^2 - 1)^3*a*b *d^3*x*arcsin(c*x)/c^2 + 2/105*(c^2*x^2 - 1)^2*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/81*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 - 622/2 50047*(c^2*x^2 - 1)^3*b^2*d^3*x/c^2 + 4/105*(c^2*x^2 - 1)^2*a*b*d^3*x*arcs in(c*x)/c^2 - 8/315*(c^2*x^2 - 1)*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/81*(c^2* x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3 - 2/441*(c^2*x^2 - 1)^3*sqrt(-c^ 2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 + 15224/10418625*(c^2*x^2 - 1)^2*b^2*d^ 3*x/c^2 - 16/315*(c^2*x^2 - 1)*a*b*d^3*x*arcsin(c*x)/c^2 + 16/315*b^2*d^3* x*arcsin(c*x)^2/c^2 - 2/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3 + 4/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 + 1155 04/31255875*(c^2*x^2 - 1)*b^2*d^3*x/c^2 + 32/315*a*b*d^3*x*arcsin(c*x)/c^2 + 4/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3 + 16/945*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*arcsin(c*x)/c^3 - 3406208/31255875*b^2*d^3*x/c^2 + 16/ 945*(-c^2*x^2 + 1)^(3/2)*a*b*d^3/c^3 + 32/315*sqrt(-c^2*x^2 + 1)*b^2*d^3*a rcsin(c*x)/c^3 + 32/315*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]