Integrand size = 27, antiderivative size = 384 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {79 b^2 d^3 x^2}{5120 c^2}-\frac {79 b^2 d^3 x^4}{15360}+\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {1}{500} b^2 c^6 d^3 x^{10}+\frac {79 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2560 c^3}+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3840 c}-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {79 d^3 (a+b \arcsin (c x))^2}{5120 c^4}+\frac {1}{40} d^3 x^4 (a+b \arcsin (c x))^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 \]
-79/5120*b^2*d^3*x^2/c^2-79/15360*b^2*d^3*x^4+401/28800*b^2*c^2*d^3*x^6-57 /6400*b^2*c^4*d^3*x^8+1/500*b^2*c^6*d^3*x^10-1/32*b*c*d^3*x^5*(-c^2*x^2+1) ^(3/2)*(a+b*arcsin(c*x))-1/50*b*c*d^3*x^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c *x))-79/5120*d^3*(a+b*arcsin(c*x))^2/c^4+1/40*d^3*x^4*(a+b*arcsin(c*x))^2+ 1/20*d^3*x^4*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+3/40*d^3*x^4*(-c^2*x^2+1)^2* (a+b*arcsin(c*x))^2+1/10*d^3*x^4*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2+79/256 0*b*d^3*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+79/3840*b*d^3*x^3*(a+b* arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c-31/960*b*c*d^3*x^5*(a+b*arcsin(c*x))*(-c ^2*x^2+1)^(1/2)
Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.75 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {d^3 \left (c x \left (28800 a^2 c^3 x^3 \left (-10+20 c^2 x^2-15 c^4 x^4+4 c^6 x^6\right )+30 a b \sqrt {1-c^2 x^2} \left (-1185-790 c^2 x^2+3208 c^4 x^4-2736 c^6 x^6+768 c^8 x^8\right )+b^2 \left (17775 c x+5925 c^3 x^3-16040 c^5 x^5+10260 c^7 x^7-2304 c^9 x^9\right )\right )+30 b \left (b c x \sqrt {1-c^2 x^2} \left (-1185-790 c^2 x^2+3208 c^4 x^4-2736 c^6 x^6+768 c^8 x^8\right )+15 a \left (79-1280 c^4 x^4+2560 c^6 x^6-1920 c^8 x^8+512 c^{10} x^{10}\right )\right ) \arcsin (c x)+225 b^2 \left (79-1280 c^4 x^4+2560 c^6 x^6-1920 c^8 x^8+512 c^{10} x^{10}\right ) \arcsin (c x)^2\right )}{1152000 c^4} \]
-1/1152000*(d^3*(c*x*(28800*a^2*c^3*x^3*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4 *c^6*x^6) + 30*a*b*Sqrt[1 - c^2*x^2]*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 2736*c^6*x^6 + 768*c^8*x^8) + b^2*(17775*c*x + 5925*c^3*x^3 - 16040*c^5*x ^5 + 10260*c^7*x^7 - 2304*c^9*x^9)) + 30*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 2736*c^6*x^6 + 768*c^8*x^8) + 15*a*(79 - 1 280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10))*ArcSin[c*x] + 225*b^2*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10)* ArcSin[c*x]^2))/c^4
Leaf count is larger than twice the leaf count of optimal. \(916\) vs. \(2(384)=768\).
Time = 3.51 (sec) , antiderivative size = 916, normalized size of antiderivative = 2.39, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {5202, 27, 5202, 243, 49, 2009, 5202, 244, 2009, 5138, 5198, 15, 5210, 15, 5210, 15, 5152}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{5} b c d^3 \int x^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {3}{5} d \int d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} b c d^3 \int x^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx+\frac {3}{5} d^3 \int x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{10} b c \int x^5 \left (1-c^2 x^2\right )^2dx+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )+\frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{20} b c \int x^4 \left (1-c^2 x^2\right )^2dx^2+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )+\frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {1}{20} b c \int \left (c^4 x^8-2 c^2 x^6+x^4\right )dx^2+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int x^5 \left (1-c^2 x^2\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int x^5 \left (1-c^2 x^2\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int \left (x^5-c^2 x^7\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int \left (x^5-c^2 x^7\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{10} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {b \int x^3dx}{4 c}+\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {b \int x^3dx}{4 c}+\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {b \int x^3dx}{4 c}+\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {b \int x^3dx}{4 c}+\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{10} x^5 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{10} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b \int xdx}{2 c}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b \int xdx}{2 c}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b \int xdx}{2 c}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b \int xdx}{2 c}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{10} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{10} d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {(a+b \arcsin (c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) x^5-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {(a+b \arcsin (c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^5+\frac {1}{6} \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {(a+b \arcsin (c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {b x^4}{16 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^3}{4 c^2}+\frac {3 \left (\frac {b x^2}{4 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x}{2 c^2}+\frac {(a+b \arcsin (c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )\right )\) |
(d^3*x^4*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/10 - (b*c*d^3*(-1/20*(b*c* (x^6/3 - (c^2*x^8)/2 + (c^4*x^10)/5)) + (x^5*(1 - c^2*x^2)^(5/2)*(a + b*Ar cSin[c*x]))/10 + (-1/8*(b*c*(x^6/6 - (c^2*x^8)/8)) + (x^5*(1 - c^2*x^2)^(3 /2)*(a + b*ArcSin[c*x]))/8 + (3*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 - c^2*x^2]* (a + b*ArcSin[c*x]))/6 + ((b*x^4)/(16*c) - (x^3*Sqrt[1 - c^2*x^2]*(a + b*A rcSin[c*x]))/(4*c^2) + (3*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x^2]*(a + b*Arc Sin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8)/2)) /5 + (3*d^3*((x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/8 - (b*c*(-1/8*(b *c*(x^6/6 - (c^2*x^8)/8)) + (x^5*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/ 8 + (3*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/6 + ((b*x^4)/(16*c) - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(4*c^2) + (3 *((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8))/4 + ((x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/6 - (b*c*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/6 + ((b*x^4)/(16*c) - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcS in[c*x]))/(4*c^2) + (3*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin [c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/3 + ((x^4 *(a + b*ArcSin[c*x])^2)/4 - (b*c*((b*x^4)/(16*c) - (x^3*Sqrt[1 - c^2*x^2]* (a + b*ArcSin[c*x]))/(4*c^2) + (3*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*c^2))...
3.2.75.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*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]
Time = 0.24 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.35
| method | result | size |
| parts | \(-d^{3} a^{2} \left (\frac {1}{10} c^{6} x^{10}-\frac {3}{8} c^{4} x^{8}+\frac {1}{2} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {49 \arcsin \left (c x \right )^{2}}{5120}-\frac {7 \left (c^{2} x^{2}-1\right )^{4}}{6400}+\frac {49 \left (c^{2} x^{2}-1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}-1\right )^{2}}{15360}+\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\arcsin \left (c x \right ) \left (128 c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}-656 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+1368 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-1490 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+965 c x \sqrt {-c^{2} x^{2}+1}+315 \arcsin \left (c x \right )\right )}{6400}-\frac {\left (c^{2} x^{2}-1\right )^{5}}{500}\right )}{c^{4}}-\frac {2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arcsin \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}+\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}-\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}+\frac {79 \arcsin \left (c x \right )}{5120}\right )}{c^{4}}\) | \(518\) |
| derivativedivides | \(\frac {-d^{3} a^{2} \left (\frac {1}{10} c^{10} x^{10}-\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {49 \arcsin \left (c x \right )^{2}}{5120}-\frac {7 \left (c^{2} x^{2}-1\right )^{4}}{6400}+\frac {49 \left (c^{2} x^{2}-1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}-1\right )^{2}}{15360}+\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\arcsin \left (c x \right ) \left (128 c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}-656 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+1368 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-1490 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+965 c x \sqrt {-c^{2} x^{2}+1}+315 \arcsin \left (c x \right )\right )}{6400}-\frac {\left (c^{2} x^{2}-1\right )^{5}}{500}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arcsin \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}+\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}-\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}+\frac {79 \arcsin \left (c x \right )}{5120}\right )}{c^{4}}\) | \(519\) |
| default | \(\frac {-d^{3} a^{2} \left (\frac {1}{10} c^{10} x^{10}-\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {49 \arcsin \left (c x \right )^{2}}{5120}-\frac {7 \left (c^{2} x^{2}-1\right )^{4}}{6400}+\frac {49 \left (c^{2} x^{2}-1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}-1\right )^{2}}{15360}+\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\arcsin \left (c x \right ) \left (128 c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}-656 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+1368 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-1490 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+965 c x \sqrt {-c^{2} x^{2}+1}+315 \arcsin \left (c x \right )\right )}{6400}-\frac {\left (c^{2} x^{2}-1\right )^{5}}{500}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arcsin \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}+\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}-\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}+\frac {79 \arcsin \left (c x \right )}{5120}\right )}{c^{4}}\) | \(519\) |
-d^3*a^2*(1/10*c^6*x^10-3/8*c^4*x^8+1/2*c^2*x^6-1/4*x^4)-d^3*b^2/c^4*(1/8* arcsin(c*x)^2*(c^2*x^2-1)^4-1/1536*arcsin(c*x)*(-48*c^7*x^7*(-c^2*x^2+1)^( 1/2)+200*c^5*x^5*(-c^2*x^2+1)^(1/2)-326*c^3*x^3*(-c^2*x^2+1)^(1/2)+279*c*x *(-c^2*x^2+1)^(1/2)+105*arcsin(c*x))+49/5120*arcsin(c*x)^2-7/6400*(c^2*x^2 -1)^4+49/28800*(c^2*x^2-1)^3-49/15360*(c^2*x^2-1)^2+49/5120*c^2*x^2-49/512 0+1/10*arcsin(c*x)^2*(c^2*x^2-1)^5+1/6400*arcsin(c*x)*(128*c^9*x^9*(-c^2*x ^2+1)^(1/2)-656*c^7*x^7*(-c^2*x^2+1)^(1/2)+1368*c^5*x^5*(-c^2*x^2+1)^(1/2) -1490*c^3*x^3*(-c^2*x^2+1)^(1/2)+965*c*x*(-c^2*x^2+1)^(1/2)+315*arcsin(c*x ))-1/500*(c^2*x^2-1)^5)-2*d^3*a*b/c^4*(1/10*arcsin(c*x)*c^10*x^10-3/8*arcs in(c*x)*c^8*x^8+1/2*arcsin(c*x)*c^6*x^6-1/4*c^4*x^4*arcsin(c*x)+1/100*c^9* x^9*(-c^2*x^2+1)^(1/2)-57/1600*c^7*x^7*(-c^2*x^2+1)^(1/2)+401/9600*c^5*x^5 *(-c^2*x^2+1)^(1/2)-79/7680*c^3*x^3*(-c^2*x^2+1)^(1/2)-79/5120*c*x*(-c^2*x ^2+1)^(1/2)+79/5120*arcsin(c*x))
Time = 0.28 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.03 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {2304 \, {\left (50 \, a^{2} - b^{2}\right )} c^{10} d^{3} x^{10} - 540 \, {\left (800 \, a^{2} - 19 \, b^{2}\right )} c^{8} d^{3} x^{8} + 40 \, {\left (14400 \, a^{2} - 401 \, b^{2}\right )} c^{6} d^{3} x^{6} - 75 \, {\left (3840 \, a^{2} - 79 \, b^{2}\right )} c^{4} d^{3} x^{4} + 17775 \, b^{2} c^{2} d^{3} x^{2} + 225 \, {\left (512 \, b^{2} c^{10} d^{3} x^{10} - 1920 \, b^{2} c^{8} d^{3} x^{8} + 2560 \, b^{2} c^{6} d^{3} x^{6} - 1280 \, b^{2} c^{4} d^{3} x^{4} + 79 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (512 \, a b c^{10} d^{3} x^{10} - 1920 \, a b c^{8} d^{3} x^{8} + 2560 \, a b c^{6} d^{3} x^{6} - 1280 \, a b c^{4} d^{3} x^{4} + 79 \, a b d^{3}\right )} \arcsin \left (c x\right ) + 30 \, {\left (768 \, a b c^{9} d^{3} x^{9} - 2736 \, a b c^{7} d^{3} x^{7} + 3208 \, a b c^{5} d^{3} x^{5} - 790 \, a b c^{3} d^{3} x^{3} - 1185 \, a b c d^{3} x + {\left (768 \, b^{2} c^{9} d^{3} x^{9} - 2736 \, b^{2} c^{7} d^{3} x^{7} + 3208 \, b^{2} c^{5} d^{3} x^{5} - 790 \, b^{2} c^{3} d^{3} x^{3} - 1185 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{1152000 \, c^{4}} \]
-1/1152000*(2304*(50*a^2 - b^2)*c^10*d^3*x^10 - 540*(800*a^2 - 19*b^2)*c^8 *d^3*x^8 + 40*(14400*a^2 - 401*b^2)*c^6*d^3*x^6 - 75*(3840*a^2 - 79*b^2)*c ^4*d^3*x^4 + 17775*b^2*c^2*d^3*x^2 + 225*(512*b^2*c^10*d^3*x^10 - 1920*b^2 *c^8*d^3*x^8 + 2560*b^2*c^6*d^3*x^6 - 1280*b^2*c^4*d^3*x^4 + 79*b^2*d^3)*a rcsin(c*x)^2 + 450*(512*a*b*c^10*d^3*x^10 - 1920*a*b*c^8*d^3*x^8 + 2560*a* b*c^6*d^3*x^6 - 1280*a*b*c^4*d^3*x^4 + 79*a*b*d^3)*arcsin(c*x) + 30*(768*a *b*c^9*d^3*x^9 - 2736*a*b*c^7*d^3*x^7 + 3208*a*b*c^5*d^3*x^5 - 790*a*b*c^3 *d^3*x^3 - 1185*a*b*c*d^3*x + (768*b^2*c^9*d^3*x^9 - 2736*b^2*c^7*d^3*x^7 + 3208*b^2*c^5*d^3*x^5 - 790*b^2*c^3*d^3*x^3 - 1185*b^2*c*d^3*x)*arcsin(c* x))*sqrt(-c^2*x^2 + 1))/c^4
Time = 2.35 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.70 \[ \int x^3 \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^{10}}{10} + \frac {3 a^{2} c^{4} d^{3} x^{8}}{8} - \frac {a^{2} c^{2} d^{3} x^{6}}{2} + \frac {a^{2} d^{3} x^{4}}{4} - \frac {a b c^{6} d^{3} x^{10} \operatorname {asin}{\left (c x \right )}}{5} - \frac {a b c^{5} d^{3} x^{9} \sqrt {- c^{2} x^{2} + 1}}{50} + \frac {3 a b c^{4} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{4} + \frac {57 a b c^{3} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{800} - a b c^{2} d^{3} x^{6} \operatorname {asin}{\left (c x \right )} - \frac {401 a b c d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{4800} + \frac {a b d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {79 a b d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{3840 c} + \frac {79 a b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{2560 c^{3}} - \frac {79 a b d^{3} \operatorname {asin}{\left (c x \right )}}{2560 c^{4}} - \frac {b^{2} c^{6} d^{3} x^{10} \operatorname {asin}^{2}{\left (c x \right )}}{10} + \frac {b^{2} c^{6} d^{3} x^{10}}{500} - \frac {b^{2} c^{5} d^{3} x^{9} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{50} + \frac {3 b^{2} c^{4} d^{3} x^{8} \operatorname {asin}^{2}{\left (c x \right )}}{8} - \frac {57 b^{2} c^{4} d^{3} x^{8}}{6400} + \frac {57 b^{2} c^{3} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{800} - \frac {b^{2} c^{2} d^{3} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{2} + \frac {401 b^{2} c^{2} d^{3} x^{6}}{28800} - \frac {401 b^{2} c d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{4800} + \frac {b^{2} d^{3} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {79 b^{2} d^{3} x^{4}}{15360} + \frac {79 b^{2} d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3840 c} - \frac {79 b^{2} d^{3} x^{2}}{5120 c^{2}} + \frac {79 b^{2} d^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2560 c^{3}} - \frac {79 b^{2} d^{3} \operatorname {asin}^{2}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**6*d**3*x**10/10 + 3*a**2*c**4*d**3*x**8/8 - a**2*c**2* d**3*x**6/2 + a**2*d**3*x**4/4 - a*b*c**6*d**3*x**10*asin(c*x)/5 - a*b*c** 5*d**3*x**9*sqrt(-c**2*x**2 + 1)/50 + 3*a*b*c**4*d**3*x**8*asin(c*x)/4 + 5 7*a*b*c**3*d**3*x**7*sqrt(-c**2*x**2 + 1)/800 - a*b*c**2*d**3*x**6*asin(c* x) - 401*a*b*c*d**3*x**5*sqrt(-c**2*x**2 + 1)/4800 + a*b*d**3*x**4*asin(c* x)/2 + 79*a*b*d**3*x**3*sqrt(-c**2*x**2 + 1)/(3840*c) + 79*a*b*d**3*x*sqrt (-c**2*x**2 + 1)/(2560*c**3) - 79*a*b*d**3*asin(c*x)/(2560*c**4) - b**2*c* *6*d**3*x**10*asin(c*x)**2/10 + b**2*c**6*d**3*x**10/500 - b**2*c**5*d**3* x**9*sqrt(-c**2*x**2 + 1)*asin(c*x)/50 + 3*b**2*c**4*d**3*x**8*asin(c*x)** 2/8 - 57*b**2*c**4*d**3*x**8/6400 + 57*b**2*c**3*d**3*x**7*sqrt(-c**2*x**2 + 1)*asin(c*x)/800 - b**2*c**2*d**3*x**6*asin(c*x)**2/2 + 401*b**2*c**2*d **3*x**6/28800 - 401*b**2*c*d**3*x**5*sqrt(-c**2*x**2 + 1)*asin(c*x)/4800 + b**2*d**3*x**4*asin(c*x)**2/4 - 79*b**2*d**3*x**4/15360 + 79*b**2*d**3*x **3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3840*c) - 79*b**2*d**3*x**2/(5120*c**2 ) + 79*b**2*d**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2560*c**3) - 79*b**2*d* *3*asin(c*x)**2/(5120*c**4), Ne(c, 0)), (a**2*d**3*x**4/4, True))
\[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
-1/10*a^2*c^6*d^3*x^10 + 3/8*a^2*c^4*d^3*x^8 - 1/2*a^2*c^2*d^3*x^6 - 1/640 0*(1280*x^10*arcsin(c*x) + (128*sqrt(-c^2*x^2 + 1)*x^9/c^2 + 144*sqrt(-c^2 *x^2 + 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/c^6 + 210*sqrt(-c^2*x^2 + 1 )*x^3/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsin(c*x)/c^11)*c)*a*b*c ^6*d^3 + 1/512*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56* sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2 *x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9)*c)*a*b*c^4*d^3 + 1/4*a^2*d^3*x^4 - 1/48*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^ 2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b* c^2*d^3 + 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt (-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*a*b*d^3 - 1/40*(4*b^2*c^6*d^3 *x^10 - 15*b^2*c^4*d^3*x^8 + 20*b^2*c^2*d^3*x^6 - 10*b^2*d^3*x^4)*arctan2( c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/20*(4*b^2*c^7*d^3*x^10 - 15*b^2*c^5*d^3*x^8 + 20*b^2*c^3*d^3*x^6 - 10*b^2*c*d^3*x^4)*sqrt(c*x + 1 )*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)
Time = 0.34 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.64 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=-\frac {1}{10} \, a^{2} c^{6} d^{3} x^{10} + \frac {3}{8} \, a^{2} c^{4} d^{3} x^{8} - \frac {1}{2} \, a^{2} c^{2} d^{3} x^{6} + \frac {1}{4} \, a^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{50 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{10 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{50 \, c^{3}} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{800 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} a b d^{3} \arcsin \left (c x\right )}{5 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{8 \, c^{4}} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{800 \, c^{3}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{4800 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b^{2} d^{3}}{500 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{4800 \, c^{3}} + \frac {49 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} x \arcsin \left (c x\right )}{3840 \, c^{3}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3}}{6400 \, c^{4}} + \frac {49 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3} x}{3840 \, c^{3}} + \frac {49 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{2560 \, c^{3}} - \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3}}{28800 \, c^{4}} + \frac {49 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{2560 \, c^{3}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3}}{15360 \, c^{4}} + \frac {49 \, b^{2} d^{3} \arcsin \left (c x\right )^{2}}{5120 \, c^{4}} - \frac {49 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3}}{5120 \, c^{4}} + \frac {49 \, a b d^{3} \arcsin \left (c x\right )}{2560 \, c^{4}} - \frac {232981 \, b^{2} d^{3}}{36864000 \, c^{4}} \]
-1/10*a^2*c^6*d^3*x^10 + 3/8*a^2*c^4*d^3*x^8 - 1/2*a^2*c^2*d^3*x^6 + 1/4*a ^2*d^3*x^4 - 1/50*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x) /c^3 - 1/10*(c^2*x^2 - 1)^5*b^2*d^3*arcsin(c*x)^2/c^4 - 1/50*(c^2*x^2 - 1) ^4*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c^3 - 7/800*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 - 1/5*(c^2*x^2 - 1)^5*a*b*d^3*arcsin(c*x)/c ^4 - 1/8*(c^2*x^2 - 1)^4*b^2*d^3*arcsin(c*x)^2/c^4 - 7/800*(c^2*x^2 - 1)^3 *sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c^3 + 49/4800*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 + 1/500*(c^2*x^2 - 1)^5*b^2*d^3/c^4 - 1/4*( c^2*x^2 - 1)^4*a*b*d^3*arcsin(c*x)/c^4 + 49/4800*(c^2*x^2 - 1)^2*sqrt(-c^2 *x^2 + 1)*a*b*d^3*x/c^3 + 49/3840*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*x*arcsin(c* x)/c^3 + 7/6400*(c^2*x^2 - 1)^4*b^2*d^3/c^4 + 49/3840*(-c^2*x^2 + 1)^(3/2) *a*b*d^3*x/c^3 + 49/2560*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 - 49 /28800*(c^2*x^2 - 1)^3*b^2*d^3/c^4 + 49/2560*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/ c^3 + 49/15360*(c^2*x^2 - 1)^2*b^2*d^3/c^4 + 49/5120*b^2*d^3*arcsin(c*x)^2 /c^4 - 49/5120*(c^2*x^2 - 1)*b^2*d^3/c^4 + 49/2560*a*b*d^3*arcsin(c*x)/c^4 - 232981/36864000*b^2*d^3/c^4
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]