Integrand size = 27, antiderivative size = 302 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=-\frac {73 b^2 d^2 x^2}{3072 c^2}-\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}-\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{1536 c^3}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {73 d^2 (a+b \arcsin (c x))^2}{3072 c^4}+\frac {1}{24} d^2 x^4 (a+b \arcsin (c x))^2+\frac {1}{12} d^2 x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 \]
-73/3072*b^2*d^2*x^2/c^2-73/9216*b^2*d^2*x^4+43/3456*b^2*c^2*d^2*x^6-1/256 *b^2*c^4*d^2*x^8-1/32*b*c*d^2*x^5*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-73/ 3072*d^2*(a+b*arcsin(c*x))^2/c^4+1/24*d^2*x^4*(a+b*arcsin(c*x))^2+1/12*d^2 *x^4*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2+1/8*d^2*x^4*(-c^2*x^2+1)^2*(a+b*arcs in(c*x))^2+73/1536*b*d^2*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+73/230 4*b*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c-25/576*b*c*d^2*x^5*(a+b *arcsin(c*x))*(-c^2*x^2+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (6-8 c^2 x^2+3 c^4 x^4\right )-b^2 c x \left (657+219 c^2 x^2-344 c^4 x^4+108 c^6 x^6\right )+6 a b \sqrt {1-c^2 x^2} \left (219+146 c^2 x^2-344 c^4 x^4+144 c^6 x^6\right )\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (219+146 c^2 x^2-344 c^4 x^4+144 c^6 x^6\right )+3 a \left (-73+768 c^4 x^4-1024 c^6 x^6+384 c^8 x^8\right )\right ) \arcsin (c x)+9 b^2 \left (-73+768 c^4 x^4-1024 c^6 x^6+384 c^8 x^8\right ) \arcsin (c x)^2\right )}{27648 c^4} \]
(d^2*(c*x*(1152*a^2*c^3*x^3*(6 - 8*c^2*x^2 + 3*c^4*x^4) - b^2*c*x*(657 + 2 19*c^2*x^2 - 344*c^4*x^4 + 108*c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(219 + 1 46*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(2 19 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6) + 3*a*(-73 + 768*c^4*x^4 - 1 024*c^6*x^6 + 384*c^8*x^8))*ArcSin[c*x] + 9*b^2*(-73 + 768*c^4*x^4 - 1024* c^6*x^6 + 384*c^8*x^8)*ArcSin[c*x]^2))/(27648*c^4)
Time = 2.20 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.97, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5202, 27, 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.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{4} b c d^2 \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} d \int d x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} b c d^2 \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} d^2 \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int x^5 \left (1-c^2 x^2\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{8} b c \int \left (x^5-c^2 x^7\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d^2 \left (-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} \int x^3 (a+b \arcsin (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle -\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} d^2 \left (\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )\right )-\frac {1}{4} b c d^2 \left (\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )-\frac {1}{36} b c x^6\right )-\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )\) |
(d^2*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/8 - (b*c*d^2*(-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*Ar cSin[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8))/4 + (d^2*((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)))/2 )/3))/2
3.2.66.3.1 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[((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 = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.40
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{8} c^{4} x^{8}-\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {55 \arcsin \left (c x \right )^{2}}{3072}-\frac {11 \left (c^{2} x^{2}-1\right )^{3}}{3456}+\frac {55 \left (c^{2} x^{2}-1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}+\frac {55}{3072}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}\right )}{c^{4}}+\frac {2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}+\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}-\frac {73 \arcsin \left (c x \right )}{3072}\right )}{c^{4}}\) | \(423\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {55 \arcsin \left (c x \right )^{2}}{3072}-\frac {11 \left (c^{2} x^{2}-1\right )^{3}}{3456}+\frac {55 \left (c^{2} x^{2}-1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}+\frac {55}{3072}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}+\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}-\frac {73 \arcsin \left (c x \right )}{3072}\right )}{c^{4}}\) | \(424\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arcsin \left (c x \right ) \left (8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+33 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{144}-\frac {55 \arcsin \left (c x \right )^{2}}{3072}-\frac {11 \left (c^{2} x^{2}-1\right )^{3}}{3456}+\frac {55 \left (c^{2} x^{2}-1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}+\frac {55}{3072}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}+\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}-\frac {73 \arcsin \left (c x \right )}{3072}\right )}{c^{4}}\) | \(424\) |
d^2*a^2*(1/8*c^4*x^8-1/3*c^2*x^6+1/4*x^4)+d^2*b^2/c^4*(1/6*arcsin(c*x)^2*( c^2*x^2-1)^3+1/144*arcsin(c*x)*(8*c^5*x^5*(-c^2*x^2+1)^(1/2)-26*c^3*x^3*(- c^2*x^2+1)^(1/2)+33*c*x*(-c^2*x^2+1)^(1/2)+15*arcsin(c*x))-55/3072*arcsin( c*x)^2-11/3456*(c^2*x^2-1)^3+55/9216*(c^2*x^2-1)^2-55/3072*c^2*x^2+55/3072 +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)+27 9*c*x*(-c^2*x^2+1)^(1/2)+105*arcsin(c*x))-1/256*(c^2*x^2-1)^4)+2*d^2*a*b/c ^4*(1/8*arcsin(c*x)*c^8*x^8-1/3*arcsin(c*x)*c^6*x^6+1/4*c^4*x^4*arcsin(c*x )+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-43/1152*c^5*x^5*(-c^2*x^2+1)^(1/2)+73/46 08*c^3*x^3*(-c^2*x^2+1)^(1/2)+73/3072*c*x*(-c^2*x^2+1)^(1/2)-73/3072*arcsi n(c*x))
Time = 0.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.06 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {108 \, {\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{2} x^{8} - 8 \, {\left (1152 \, a^{2} - 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (2304 \, a^{2} - 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \, {\left (384 \, b^{2} c^{8} d^{2} x^{8} - 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (384 \, a b c^{8} d^{2} x^{8} - 1024 \, a b c^{6} d^{2} x^{6} + 768 \, a b c^{4} d^{2} x^{4} - 73 \, a b d^{2}\right )} \arcsin \left (c x\right ) + 6 \, {\left (144 \, a b c^{7} d^{2} x^{7} - 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} + 219 \, a b c d^{2} x + {\left (144 \, b^{2} c^{7} d^{2} x^{7} - 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} + 219 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27648 \, c^{4}} \]
1/27648*(108*(32*a^2 - b^2)*c^8*d^2*x^8 - 8*(1152*a^2 - 43*b^2)*c^6*d^2*x^ 6 + 3*(2304*a^2 - 73*b^2)*c^4*d^2*x^4 - 657*b^2*c^2*d^2*x^2 + 9*(384*b^2*c ^8*d^2*x^8 - 1024*b^2*c^6*d^2*x^6 + 768*b^2*c^4*d^2*x^4 - 73*b^2*d^2)*arcs in(c*x)^2 + 18*(384*a*b*c^8*d^2*x^8 - 1024*a*b*c^6*d^2*x^6 + 768*a*b*c^4*d ^2*x^4 - 73*a*b*d^2)*arcsin(c*x) + 6*(144*a*b*c^7*d^2*x^7 - 344*a*b*c^5*d^ 2*x^5 + 146*a*b*c^3*d^2*x^3 + 219*a*b*c*d^2*x + (144*b^2*c^7*d^2*x^7 - 344 *b^2*c^5*d^2*x^5 + 146*b^2*c^3*d^2*x^3 + 219*b^2*c*d^2*x)*arcsin(c*x))*sqr t(-c^2*x^2 + 1))/c^4
Time = 1.24 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.71 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{8}}{8} - \frac {a^{2} c^{2} d^{2} x^{6}}{3} + \frac {a^{2} d^{2} x^{4}}{4} + \frac {a b c^{4} d^{2} x^{8} \operatorname {asin}{\left (c x \right )}}{4} + \frac {a b c^{3} d^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{32} - \frac {2 a b c^{2} d^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{3} - \frac {43 a b c d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{576} + \frac {a b d^{2} x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {73 a b d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{2304 c} + \frac {73 a b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{1536 c^{3}} - \frac {73 a b d^{2} \operatorname {asin}{\left (c x \right )}}{1536 c^{4}} + \frac {b^{2} c^{4} d^{2} x^{8} \operatorname {asin}^{2}{\left (c x \right )}}{8} - \frac {b^{2} c^{4} d^{2} x^{8}}{256} + \frac {b^{2} c^{3} d^{2} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{32} - \frac {b^{2} c^{2} d^{2} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{3} + \frac {43 b^{2} c^{2} d^{2} x^{6}}{3456} - \frac {43 b^{2} c d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{576} + \frac {b^{2} d^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {73 b^{2} d^{2} x^{4}}{9216} + \frac {73 b^{2} d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2304 c} - \frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}} + \frac {73 b^{2} d^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1536 c^{3}} - \frac {73 b^{2} d^{2} \operatorname {asin}^{2}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**4*d**2*x**8/8 - a**2*c**2*d**2*x**6/3 + a**2*d**2*x**4/ 4 + a*b*c**4*d**2*x**8*asin(c*x)/4 + a*b*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)/32 - 2*a*b*c**2*d**2*x**6*asin(c*x)/3 - 43*a*b*c*d**2*x**5*sqrt(-c**2*x **2 + 1)/576 + a*b*d**2*x**4*asin(c*x)/2 + 73*a*b*d**2*x**3*sqrt(-c**2*x** 2 + 1)/(2304*c) + 73*a*b*d**2*x*sqrt(-c**2*x**2 + 1)/(1536*c**3) - 73*a*b* d**2*asin(c*x)/(1536*c**4) + b**2*c**4*d**2*x**8*asin(c*x)**2/8 - b**2*c** 4*d**2*x**8/256 + b**2*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)*asin(c*x)/32 - b**2*c**2*d**2*x**6*asin(c*x)**2/3 + 43*b**2*c**2*d**2*x**6/3456 - 43*b**2 *c*d**2*x**5*sqrt(-c**2*x**2 + 1)*asin(c*x)/576 + b**2*d**2*x**4*asin(c*x) **2/4 - 73*b**2*d**2*x**4/9216 + 73*b**2*d**2*x**3*sqrt(-c**2*x**2 + 1)*as in(c*x)/(2304*c) - 73*b**2*d**2*x**2/(3072*c**2) + 73*b**2*d**2*x*sqrt(-c* *2*x**2 + 1)*asin(c*x)/(1536*c**3) - 73*b**2*d**2*asin(c*x)**2/(3072*c**4) , Ne(c, 0)), (a**2*d**2*x**4/4, True))
\[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
1/8*a^2*c^4*d^2*x^8 - 1/3*a^2*c^2*d^2*x^6 + 1/1536*(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^2 + 1/4*a^2*d^2*x^4 - 1/72*(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^2 + 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^2 + 1/24*(3*b^2*c^4*d^2*x^8 - 8*b^2*c^2*d^2*x^6 + 6*b^2*d^2*x ^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/12*(3*b^2*c ^5*d^2*x^8 - 8*b^2*c^3*d^2*x^6 + 6*b^2*c*d^2*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 = 522, normalized size of antiderivative = 1.73 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {1}{8} \, a^{2} c^{4} d^{2} x^{8} - \frac {1}{3} \, a^{2} c^{2} d^{2} x^{6} + \frac {1}{4} \, a^{2} d^{2} x^{4} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{8 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{32 \, c^{3}} + \frac {11 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{576 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{2} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2} \arcsin \left (c x\right )^{2}}{6 \, c^{4}} + \frac {11 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{576 \, c^{3}} + \frac {55 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} x \arcsin \left (c x\right )}{2304 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{2}}{256 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} a b d^{2} \arcsin \left (c x\right )}{3 \, c^{4}} + \frac {55 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2} x}{2304 \, c^{3}} + \frac {55 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right )}{1536 \, c^{3}} - \frac {11 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{2}}{3456 \, c^{4}} + \frac {55 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{1536 \, c^{3}} + \frac {55 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2}}{9216 \, c^{4}} + \frac {55 \, b^{2} d^{2} \arcsin \left (c x\right )^{2}}{3072 \, c^{4}} - \frac {55 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2}}{3072 \, c^{4}} + \frac {55 \, a b d^{2} \arcsin \left (c x\right )}{1536 \, c^{4}} - \frac {9835 \, b^{2} d^{2}}{884736 \, c^{4}} \]
1/8*a^2*c^4*d^2*x^8 - 1/3*a^2*c^2*d^2*x^6 + 1/4*a^2*d^2*x^4 + 1/32*(c^2*x^ 2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x)/c^3 + 1/8*(c^2*x^2 - 1)^ 4*b^2*d^2*arcsin(c*x)^2/c^4 + 1/32*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b* d^2*x/c^3 + 11/576*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*x )/c^3 + 1/4*(c^2*x^2 - 1)^4*a*b*d^2*arcsin(c*x)/c^4 + 1/6*(c^2*x^2 - 1)^3* b^2*d^2*arcsin(c*x)^2/c^4 + 11/576*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b* d^2*x/c^3 + 55/2304*(-c^2*x^2 + 1)^(3/2)*b^2*d^2*x*arcsin(c*x)/c^3 - 1/256 *(c^2*x^2 - 1)^4*b^2*d^2/c^4 + 1/3*(c^2*x^2 - 1)^3*a*b*d^2*arcsin(c*x)/c^4 + 55/2304*(-c^2*x^2 + 1)^(3/2)*a*b*d^2*x/c^3 + 55/1536*sqrt(-c^2*x^2 + 1) *b^2*d^2*x*arcsin(c*x)/c^3 - 11/3456*(c^2*x^2 - 1)^3*b^2*d^2/c^4 + 55/1536 *sqrt(-c^2*x^2 + 1)*a*b*d^2*x/c^3 + 55/9216*(c^2*x^2 - 1)^2*b^2*d^2/c^4 + 55/3072*b^2*d^2*arcsin(c*x)^2/c^4 - 55/3072*(c^2*x^2 - 1)*b^2*d^2/c^4 + 55 /1536*a*b*d^2*arcsin(c*x)/c^4 - 9835/884736*b^2*d^2/c^4
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]