Integrand size = 26, antiderivative size = 438 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {245 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1152}-\frac {65 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}-\frac {1}{108} b^2 d^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {115 b^2 d^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \]
5/24*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2+1/6*x*(-c^2*d*x^2+d)^(5/ 2)*(a+b*arcsin(c*x))^2-245/1152*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-65/1728*b^2 *d^2*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)-1/108*b^2*d^2*x*(-c^2*x^2+1)^2*(- c^2*d*x^2+d)^(1/2)+5/48*b*d^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*(-c^2*d *x^2+d)^(1/2)/c+1/18*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*(-c^2*d*x^ 2+d)^(1/2)/c+5/16*d^2*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+115/1152* b^2*d^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2 *x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/48*d^2*(a +b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)
Time = 0.79 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.68 \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (360 a^3+b^3 c x \sqrt {1-c^2 x^2} \left (-897+194 c^2 x^2-32 c^4 x^4\right )-24 a b^2 c^2 x^2 \left (99-39 c^2 x^2+8 c^4 x^4\right )+72 a^2 b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+3 b \left (360 a^2+48 a b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )+b^2 \left (299-792 c^2 x^2+312 c^4 x^4-64 c^6 x^6\right )\right ) \arcsin (c x)+72 b^2 \left (15 a+b c x \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )\right ) \arcsin (c x)^2+360 b^3 \arcsin (c x)^3\right )}{3456 b c \sqrt {1-c^2 x^2}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(360*a^3 + b^3*c*x*Sqrt[1 - c^2*x^2]*(-897 + 194* c^2*x^2 - 32*c^4*x^4) - 24*a*b^2*c^2*x^2*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 7 2*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 3*b*(360*a^2 + 48*a*b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + b^2*(299 - 792*c^2*x^2 + 312*c^4*x^4 - 64*c^6*x^6))*ArcSin[c*x] + 72*b^2*(15*a + b*c *x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x]^2 + 360*b^ 3*ArcSin[c*x]^3))/(3456*b*c*Sqrt[1 - c^2*x^2])
Time = 1.51 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5158, 5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 211, 223}
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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \int x (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \int \left (1-c^2 x^2\right )^{5/2}dx}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2+\frac {5}{6} d \left (\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}\right )\right )\) |
(x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/6 - (b*c*d^2*Sqrt[d - c^2* d*x^2]*(-1/6*((1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x ^2)^(5/2))/6 + (5*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/6))/(6*c)))/(3*Sqrt[1 - c^2*x^2]) + (5*d*((x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*d*((x*Sqrt[d - c^2*d*x^2 ]*(a + b*ArcSin[c*x])^2)/2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/( 6*b*c*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*((x^2*(a + b*ArcSin[c* x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2))/ Sqrt[1 - c^2*x^2]))/4 - (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/4*((1 - c^2*x^2)^2* (a + b*ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 - c^2*x^2])))/6
3.3.29.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
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]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 1349, normalized size of antiderivative = 3.08
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1349\) |
| parts | \(\text {Expression too large to display}\) | \(1349\) |
1/6*x*(-c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a^2* d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) *x/(-c^2*d*x^2+d)^(1/2))+b^2*(-5/48*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1 /2)/c/(c^2*x^2-1)*arcsin(c*x)^3*d^2+1/6912*(-d*(c^2*x^2-1))^(1/2)*(-32*I*( -c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c^ 5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c* x)*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d^2/c/(c^2*x^2-1)+15/256*(-d*(c^2* x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/ 2)-2*c*x)*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)*d^2/c/(c^2*x^2-1)-1/27648*( -d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(348*I*arcsin(c *x)+576*arcsin(c*x)^2-77)*cos(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)+5/27648*(-d *(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(60*I*arcsin(c*x) +144*arcsin(c*x)^2-17)*sin(5*arcsin(c*x))*d^2/c/(c^2*x^2-1)-3/1024*(-d*(c^ 2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(44*I*arcsin(c*x)+32* arcsin(c*x)^2-19)*cos(3*arcsin(c*x))*d^2/c/(c^2*x^2-1)+9/1024*(-d*(c^2*x^2 -1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(12*I*arcsin(c*x)+16*arcsi n(c*x)^2-7)*sin(3*arcsin(c*x))*d^2/c/(c^2*x^2-1))+2*a*b*(-5/32*(-d*(c^2*x^ 2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arcsin(c*x)^2*d^2+1/2304*(-d* (c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2 *x^2+1)^(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3...
\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
\[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt (-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*arcsin(c*x)/c)*a^2 + sqrt(d)*integrate (((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*a rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x )
Exception generated. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]