Integrand size = 27, antiderivative size = 395 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=-\frac {4208 b^2 d^2 x}{99225 c^4}-\frac {2104 b^2 d^2 x^3}{297675 c^2}-\frac {526 b^2 d^2 x^5}{165375}+\frac {212 b^2 c^2 d^2 x^7}{27783}-\frac {2}{729} b^2 c^4 d^2 x^9+\frac {128 b d^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4725 c^5}+\frac {64 b d^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4725 c^3}+\frac {16 b d^2 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{1575 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{189 c^5}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{315 c^5}-\frac {20 b d^2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{441 c^5}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{81 c^5}+\frac {8}{315} d^2 x^5 (a+b \arcsin (c x))^2+\frac {4}{63} d^2 x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 \]
-4208/99225*b^2*d^2*x/c^4-2104/297675*b^2*d^2*x^3/c^2-526/165375*b^2*d^2*x ^5+212/27783*b^2*c^2*d^2*x^7-2/729*b^2*c^4*d^2*x^9+8/189*b*d^2*(-c^2*x^2+1 )^(3/2)*(a+b*arcsin(c*x))/c^5-2/315*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c *x))/c^5-20/441*b*d^2*(-c^2*x^2+1)^(7/2)*(a+b*arcsin(c*x))/c^5+2/81*b*d^2* (-c^2*x^2+1)^(9/2)*(a+b*arcsin(c*x))/c^5+8/315*d^2*x^5*(a+b*arcsin(c*x))^2 +4/63*d^2*x^5*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+1/9*d^2*x^5*(-c^2*x^2+1)^2* (a+b*arcsin(c*x))^2+128/4725*b*d^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^ 5+64/4725*b*d^2*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+16/1575*b*d^2 *x^4*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.15 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.64 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \left (99225 a^2 c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )+630 a b \sqrt {1-c^2 x^2} \left (2104+1052 c^2 x^2+789 c^4 x^4-2650 c^6 x^6+1225 c^8 x^8\right )-2 b^2 c x \left (662760+110460 c^2 x^2+49707 c^4 x^4-119250 c^6 x^6+42875 c^8 x^8\right )+630 b \left (315 a c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (2104+1052 c^2 x^2+789 c^4 x^4-2650 c^6 x^6+1225 c^8 x^8\right )\right ) \arcsin (c x)+99225 b^2 c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right ) \arcsin (c x)^2\right )}{31255875 c^5} \]
(d^2*(99225*a^2*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + 630*a*b*Sqrt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) - 2*b^2*c*x*(662760 + 110460*c^2*x^2 + 49707*c^4*x^4 - 119250*c^6*x^6 + 4 2875*c^8*x^8) + 630*b*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + b*Sq rt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c ^8*x^8))*ArcSin[c*x] + 99225*b^2*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*Ar cSin[c*x]^2))/(31255875*c^5)
Time = 2.15 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {5202, 27, 5194, 27, 1467, 2009, 5202, 5138, 5194, 27, 2009, 5210, 15, 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.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2}{9} b c d^2 \int x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {4}{9} d \int d x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{9} b c d^2 \int x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {4}{9} d^2 \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^2 \left (-b c \int -\frac {\left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )}{315 c^6}dx-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^2 \left (\frac {b \int \left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )dx}{315 c^5}-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx-\frac {2}{9} b c d^2 \left (\frac {b \int \left (35 c^8 x^8-50 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{315 c^5}-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{9} d^2 \int x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {4}{9} d^2 \left (-\frac {2}{7} b c \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{7} \int x^4 (a+b \arcsin (c x))^2dx+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \left (-b c \int -\frac {-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8}{105 c^6}dx-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c \left (\frac {b \int \left (-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{105 c^5}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}+\frac {b \int x^4dx}{5 c}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {4}{9} d^2 \left (\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )+\frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} d^2 x^5 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {4}{9} d^2 \left (\frac {1}{7} x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{7} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\right )-\frac {2}{9} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6}+\frac {2 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}+\frac {b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{315 c^5}\right )\) |
(d^2*x^5*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/9 - (2*b*c*d^2*((b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (50*c^6*x^7)/7 + (35*c^8*x^9)/9))/(315*c^5 ) - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^6) + (2*(1 - c^2*x^2)^( 7/2)*(a + b*ArcSin[c*x]))/(7*c^6) - ((1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x ]))/(9*c^6)))/9 + (4*d^2*((x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7 - (2 *b*c*((b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (15*c^6*x^7)/7))/(105*c^5) - ((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^6) + (2*(1 - c^2*x^2)^(5 /2)*(a + b*ArcSin[c*x]))/(5*c^6) - ((1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x] ))/(7*c^6)))/7 + (2*((x^5*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*((b*x^5)/(25*c ) - (x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(5*c^2) + (4*((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 - (S qrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/(3*c^2)))/(5*c^2)))/5))/7))/9
3.2.65.3.1 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[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -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_.))*(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 = 0.38 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.34
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{9} c^{4} x^{9}-\frac {2}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{2835}-\frac {16 c x}{315}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {20 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {4 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{3087}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )}{c^{5}}+\frac {2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(530\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{2835}-\frac {16 c x}{315}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {20 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {4 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{3087}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(531\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{2835}-\frac {16 c x}{315}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {20 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {4 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{3087}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{5}}\) | \(531\) |
d^2*a^2*(1/9*c^4*x^9-2/7*c^2*x^7+1/5*x^5)+d^2*b^2/c^5*(1/15*arcsin(c*x)^2* (3*c^4*x^4-10*c^2*x^2+15)*c*x+2/525*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1) ^(1/2)-2/7875*(3*c^4*x^4-10*c^2*x^2+15)*c*x-8/945*arcsin(c*x)*(c^2*x^2-1)* (-c^2*x^2+1)^(1/2)+8/2835*(c^2*x^2-3)*c*x-16/315*c*x+16/315*arcsin(c*x)*(- c^2*x^2+1)^(1/2)+2/35*arcsin(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c *x+20/441*arcsin(c*x)*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-4/3087*(5*c^6*x^6-2 1*c^4*x^4+35*c^2*x^2-35)*c*x+1/315*arcsin(c*x)^2*(35*c^8*x^8-180*c^6*x^6+3 78*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^2*a*b/c^5*(1/9*arcsin(c*x)*c^9*x^9-2/7*arcsin(c*x)*c^7*x^7+1/5*arcsin( c*x)*c^5*x^5+1/81*c^8*x^8*(-c^2*x^2+1)^(1/2)-106/3969*c^6*x^6*(-c^2*x^2+1) ^(1/2)+263/33075*c^4*x^4*(-c^2*x^2+1)^(1/2)+1052/99225*c^2*x^2*(-c^2*x^2+1 )^(1/2)+2104/99225*(-c^2*x^2+1)^(1/2))
Time = 0.26 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.85 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {42875 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{2} x^{9} - 2250 \, {\left (3969 \, a^{2} - 106 \, b^{2}\right )} c^{7} d^{2} x^{7} + 189 \, {\left (33075 \, a^{2} - 526 \, b^{2}\right )} c^{5} d^{2} x^{5} - 220920 \, b^{2} c^{3} d^{2} x^{3} - 1325520 \, b^{2} c d^{2} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{2} x^{9} - 90 \, b^{2} c^{7} d^{2} x^{7} + 63 \, b^{2} c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right )^{2} + 198450 \, {\left (35 \, a b c^{9} d^{2} x^{9} - 90 \, a b c^{7} d^{2} x^{7} + 63 \, a b c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + 630 \, {\left (1225 \, a b c^{8} d^{2} x^{8} - 2650 \, a b c^{6} d^{2} x^{6} + 789 \, a b c^{4} d^{2} x^{4} + 1052 \, a b c^{2} d^{2} x^{2} + 2104 \, a b d^{2} + {\left (1225 \, b^{2} c^{8} d^{2} x^{8} - 2650 \, b^{2} c^{6} d^{2} x^{6} + 789 \, b^{2} c^{4} d^{2} x^{4} + 1052 \, b^{2} c^{2} d^{2} x^{2} + 2104 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{31255875 \, c^{5}} \]
1/31255875*(42875*(81*a^2 - 2*b^2)*c^9*d^2*x^9 - 2250*(3969*a^2 - 106*b^2) *c^7*d^2*x^7 + 189*(33075*a^2 - 526*b^2)*c^5*d^2*x^5 - 220920*b^2*c^3*d^2* x^3 - 1325520*b^2*c*d^2*x + 99225*(35*b^2*c^9*d^2*x^9 - 90*b^2*c^7*d^2*x^7 + 63*b^2*c^5*d^2*x^5)*arcsin(c*x)^2 + 198450*(35*a*b*c^9*d^2*x^9 - 90*a*b *c^7*d^2*x^7 + 63*a*b*c^5*d^2*x^5)*arcsin(c*x) + 630*(1225*a*b*c^8*d^2*x^8 - 2650*a*b*c^6*d^2*x^6 + 789*a*b*c^4*d^2*x^4 + 1052*a*b*c^2*d^2*x^2 + 210 4*a*b*d^2 + (1225*b^2*c^8*d^2*x^8 - 2650*b^2*c^6*d^2*x^6 + 789*b^2*c^4*d^2 *x^4 + 1052*b^2*c^2*d^2*x^2 + 2104*b^2*d^2)*arcsin(c*x))*sqrt(-c^2*x^2 + 1 ))/c^5
Time = 1.63 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.43 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{9}}{9} - \frac {2 a^{2} c^{2} d^{2} x^{7}}{7} + \frac {a^{2} d^{2} x^{5}}{5} + \frac {2 a b c^{4} d^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {2 a b c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} - \frac {4 a b c^{2} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {212 a b c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} + \frac {2 a b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {526 a b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075 c} + \frac {2104 a b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {4208 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{5}} + \frac {b^{2} c^{4} d^{2} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{9} - \frac {2 b^{2} c^{4} d^{2} x^{9}}{729} + \frac {2 b^{2} c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{81} - \frac {2 b^{2} c^{2} d^{2} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {212 b^{2} c^{2} d^{2} x^{7}}{27783} - \frac {212 b^{2} c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3969} + \frac {b^{2} d^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {526 b^{2} d^{2} x^{5}}{165375} + \frac {526 b^{2} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{33075 c} - \frac {2104 b^{2} d^{2} x^{3}}{297675 c^{2}} + \frac {2104 b^{2} d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{3}} - \frac {4208 b^{2} d^{2} x}{99225 c^{4}} + \frac {4208 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**4*d**2*x**9/9 - 2*a**2*c**2*d**2*x**7/7 + a**2*d**2*x** 5/5 + 2*a*b*c**4*d**2*x**9*asin(c*x)/9 + 2*a*b*c**3*d**2*x**8*sqrt(-c**2*x **2 + 1)/81 - 4*a*b*c**2*d**2*x**7*asin(c*x)/7 - 212*a*b*c*d**2*x**6*sqrt( -c**2*x**2 + 1)/3969 + 2*a*b*d**2*x**5*asin(c*x)/5 + 526*a*b*d**2*x**4*sqr t(-c**2*x**2 + 1)/(33075*c) + 2104*a*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(992 25*c**3) + 4208*a*b*d**2*sqrt(-c**2*x**2 + 1)/(99225*c**5) + b**2*c**4*d** 2*x**9*asin(c*x)**2/9 - 2*b**2*c**4*d**2*x**9/729 + 2*b**2*c**3*d**2*x**8* sqrt(-c**2*x**2 + 1)*asin(c*x)/81 - 2*b**2*c**2*d**2*x**7*asin(c*x)**2/7 + 212*b**2*c**2*d**2*x**7/27783 - 212*b**2*c*d**2*x**6*sqrt(-c**2*x**2 + 1) *asin(c*x)/3969 + b**2*d**2*x**5*asin(c*x)**2/5 - 526*b**2*d**2*x**5/16537 5 + 526*b**2*d**2*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(33075*c) - 2104*b** 2*d**2*x**3/(297675*c**2) + 2104*b**2*d**2*x**2*sqrt(-c**2*x**2 + 1)*asin( c*x)/(99225*c**3) - 4208*b**2*d**2*x/(99225*c**4) + 4208*b**2*d**2*sqrt(-c **2*x**2 + 1)*asin(c*x)/(99225*c**5), Ne(c, 0)), (a**2*d**2*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (349) = 698\).
Time = 0.30 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.98 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{9} \, b^{2} c^{4} d^{2} x^{9} \arcsin \left (c x\right )^{2} + \frac {1}{9} \, a^{2} c^{4} d^{2} x^{9} - \frac {2}{7} \, b^{2} c^{2} d^{2} x^{7} \arcsin \left (c x\right )^{2} - \frac {2}{7} \, a^{2} c^{2} d^{2} x^{7} + \frac {1}{5} \, b^{2} d^{2} 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^{4} d^{2} + \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^{4} d^{2} + \frac {1}{5} \, a^{2} d^{2} x^{5} - \frac {4}{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^{2} d^{2} - \frac {4}{25725} \, {\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^{2} d^{2} + \frac {2}{75} \, {\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 d^{2} + \frac {2}{1125} \, {\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} d^{2} \]
1/9*b^2*c^4*d^2*x^9*arcsin(c*x)^2 + 1/9*a^2*c^4*d^2*x^9 - 2/7*b^2*c^2*d^2* x^7*arcsin(c*x)^2 - 2/7*a^2*c^2*d^2*x^7 + 1/5*b^2*d^2*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*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)*a*b*c^4*d^2 + 2/893025*(315*(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^4*d^2 + 1/5*a^2*d^2*x^5 - 4/245*(35*x^7*arcs in(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^2*d^2 - 4/25725*(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*arcsin(c*x) - (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^2*d^2 + 2/7 5*(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*d^2 + 2/1125*(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*d^2
Leaf count of result is larger than twice the leaf count of optimal. 702 vs. \(2 (349) = 698\).
Time = 0.36 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.78 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{9} \, a^{2} c^{4} d^{2} x^{9} - \frac {2}{7} \, a^{2} c^{2} d^{2} x^{7} + \frac {1}{5} \, a^{2} d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{9 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} a b d^{2} x \arcsin \left (c x\right )}{9 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{63 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} x}{729 \, c^{4}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} x \arcsin \left (c x\right )}{63 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{105 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{81 \, c^{5}} - \frac {836 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} x}{250047 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{2} x \arcsin \left (c x\right )}{105 \, c^{4}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{315 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{81 \, c^{5}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{441 \, c^{5}} + \frac {33862 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x}{10418625 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} a b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {8 \, b^{2} d^{2} x \arcsin \left (c x\right )^{2}}{315 \, c^{4}} + \frac {20 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{441 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{525 \, c^{5}} - \frac {47248 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x}{31255875 \, c^{4}} + \frac {16 \, a b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{525 \, c^{5}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} \arcsin \left (c x\right )}{945 \, c^{5}} - \frac {1493104 \, b^{2} d^{2} x}{31255875 \, c^{4}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2}}{945 \, c^{5}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{315 \, c^{5}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{315 \, c^{5}} \]
1/9*a^2*c^4*d^2*x^9 - 2/7*a^2*c^2*d^2*x^7 + 1/5*a^2*d^2*x^5 + 1/9*(c^2*x^2 - 1)^4*b^2*d^2*x*arcsin(c*x)^2/c^4 + 2/9*(c^2*x^2 - 1)^4*a*b*d^2*x*arcsin (c*x)/c^4 + 10/63*(c^2*x^2 - 1)^3*b^2*d^2*x*arcsin(c*x)^2/c^4 - 2/729*(c^2 *x^2 - 1)^4*b^2*d^2*x/c^4 + 20/63*(c^2*x^2 - 1)^3*a*b*d^2*x*arcsin(c*x)/c^ 4 + 1/105*(c^2*x^2 - 1)^2*b^2*d^2*x*arcsin(c*x)^2/c^4 + 2/81*(c^2*x^2 - 1) ^4*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^5 - 836/250047*(c^2*x^2 - 1)^3 *b^2*d^2*x/c^4 + 2/105*(c^2*x^2 - 1)^2*a*b*d^2*x*arcsin(c*x)/c^4 - 4/315*( c^2*x^2 - 1)*b^2*d^2*x*arcsin(c*x)^2/c^4 + 2/81*(c^2*x^2 - 1)^4*sqrt(-c^2* x^2 + 1)*a*b*d^2/c^5 + 20/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^2*a rcsin(c*x)/c^5 + 33862/10418625*(c^2*x^2 - 1)^2*b^2*d^2*x/c^4 - 8/315*(c^2 *x^2 - 1)*a*b*d^2*x*arcsin(c*x)/c^4 + 8/315*b^2*d^2*x*arcsin(c*x)^2/c^4 + 20/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^2/c^5 + 2/525*(c^2*x^2 - 1 )^2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^5 - 47248/31255875*(c^2*x^2 - 1)*b^2*d^2*x/c^4 + 16/315*a*b*d^2*x*arcsin(c*x)/c^4 + 2/525*(c^2*x^2 - 1) ^2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c^5 + 8/945*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*arc sin(c*x)/c^5 - 1493104/31255875*b^2*d^2*x/c^4 + 8/945*(-c^2*x^2 + 1)^(3/2) *a*b*d^2/c^5 + 16/315*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c^5 + 16/315* sqrt(-c^2*x^2 + 1)*a*b*d^2/c^5
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]