Integrand size = 29, antiderivative size = 556 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {359 b^2 d^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{384 b c^3 \sqrt {1-c^2 x^2}} \] Output:
-359/36864*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^2-1079/55296*b^2*d^2*x^3*(-c^2 *d*x^2+d)^(1/2)+209/13824*b^2*c^2*d^2*x^5*(-c^2*d*x^2+d)^(1/2)-1/256*b^2*c ^4*d^2*x^7*(-c^2*d*x^2+d)^(1/2)+359/36864*b^2*d^2*(-c^2*d*x^2+d)^(1/2)*arc sin(c*x)/c^3/(-c^2*x^2+1)^(1/2)+5/128*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b* arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-59/384*b*c*d^2*x^4*(-c^2*d*x^2+d)^(1/2)* (a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+17/144*b*c^3*d^2*x^6*(-c^2*d*x^2+d)^( 1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-1/32*b*c^5*d^2*x^8*(-c^2*d*x^2+d )^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-5/128*d^2*x*(-c^2*d*x^2+d)^(1 /2)*(a+b*arcsin(c*x))^2/c^2+5/64*d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin( c*x))^2+5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2+1/8*x^3*(-c^2* d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2+5/384*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arc sin(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 0.25 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.63 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (1440 a^3-96 a b^2 c^2 x^2 \left (-45+177 c^2 x^2-136 c^4 x^4+36 c^6 x^6\right )+288 a^2 b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )-b^3 c x \sqrt {1-c^2 x^2} \left (1077+2158 c^2 x^2-1672 c^4 x^4+432 c^6 x^6\right )+3 b \left (1440 a^2+192 a b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )+b^2 \left (359+1440 c^2 x^2-5664 c^4 x^4+4352 c^6 x^6-1152 c^8 x^8\right )\right ) \arcsin (c x)+288 b^2 \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right ) \arcsin (c x)^2+1440 b^3 \arcsin (c x)^3\right )}{110592 b c^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*(1440*a^3 - 96*a*b^2*c^2*x^2*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6) + 288*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^ 2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) - b^3*c*x*Sqrt[1 - c^2*x^2]*(1077 + 2158 *c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6) + 3*b*(1440*a^2 + 192*a*b*c*x*Sqrt[ 1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) + b^2*(359 + 1 440*c^2*x^2 - 5664*c^4*x^4 + 4352*c^6*x^6 - 1152*c^8*x^8))*ArcSin[c*x] + 2 88*b^2*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x]^2 + 1440*b^3*ArcSin[c*x]^3))/(110592*b*c^3*Sqrt[1 - c^2*x^2])
Time = 3.73 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.37, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {5202, 5192, 27, 1590, 25, 27, 363, 262, 262, 223, 5202, 5192, 27, 363, 262, 262, 223, 5198, 5138, 262, 262, 223, 5210, 5138, 262, 223, 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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx}{4 \sqrt {1-c^2 x^2}}+\frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{24 \sqrt {1-c^2 x^2}}dx+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (-\frac {\int -\frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (\frac {\int \frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (\frac {1}{8} \int \frac {x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int x (a+b \arcsin (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arcsin (c x))-\frac {1}{3} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}\) |
Input:
Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/8 - (b*c*d^2*Sqrt[d - c^ 2*d*x^2]*((x^4*(a + b*ArcSin[c*x]))/4 - (c^2*x^6*(a + b*ArcSin[c*x]))/3 + (c^4*x^8*(a + b*ArcSin[c*x]))/8 - (b*c*((-3*c^2*x^7*Sqrt[1 - c^2*x^2])/8 + ((43*x^5*Sqrt[1 - c^2*x^2])/6 + (73*(-1/4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + ( 3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c^2)))/6)/8)) /24))/(4*Sqrt[1 - c^2*x^2]) + (5*d*((x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcS in[c*x])^2)/6 - (b*c*d*Sqrt[d - c^2*d*x^2]*((x^4*(a + b*ArcSin[c*x]))/4 - (c^2*x^6*(a + b*ArcSin[c*x]))/6 - (b*c*((x^5*Sqrt[1 - c^2*x^2])/3 + (4*(-1 /4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + (3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcS in[c*x]/(2*c^3)))/(4*c^2)))/3))/12))/(3*Sqrt[1 - c^2*x^2]) + (d*((x^3*Sqrt [d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/4 - (b*c*Sqrt[d - c^2*d*x^2]*((x^4* (a + b*ArcSin[c*x]))/4 - (b*c*(-1/4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + (3*(-1/2 *(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c^2)))/4))/(2*Sqrt[1 - c^2*x^2]) + (Sqrt[d - c^2*d*x^2]*(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcS in[c*x])^2)/c^2 + (a + b*ArcSin[c*x])^3/(6*b*c^3) + (b*((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 ))/c))/(4*Sqrt[1 - c^2*x^2])))/2))/8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ (a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c ^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 ] && IGtQ[p, 0]
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]
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 1939, normalized size of antiderivative = 3.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1939\) |
| parts | \(\text {Expression too large to display}\) | \(1939\) |
Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
Output:
-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+ 5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d)^ (1/2)+5/128*a^2/c^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*(-5/384*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^ 2-1)*arcsin(c*x)^3*d^2+1/65536*(-d*(c^2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1) ^(1/2)*x^8*c^8+128*c^9*x^9+256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-16 0*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2 -88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x)*(8*I*arcsin(c*x)+32*arcsin(c*x)^2- 1)*d^2/c^3/(c^2*x^2-1)-1/6912*(-d*(c^2*x^2-1))^(1/2)*(-32*I*(-c^2*x^2+1)^( 1/2)*x^6*c^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*arcsi n(c*x)+18*arcsin(c*x)^2-1)*d^2/c^3/(c^2*x^2-1)+1/256*(-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 *arcsin(c*x)^2-1-2*I*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)+1/65536*(-d*(c^2*x^2 -1))^(1/2)*(128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*x^9-256*I*(-c^2*x^2+1 )^(1/2)*x^6*c^6-320*c^7*x^7+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5*x^5-3 2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+8*c*x)*(-8* I*arcsin(c*x)+32*arcsin(c*x)^2-1)*d^2/c^3/(c^2*x^2-1)+1/55296*(-d*(c^2*x^2 -1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(156*I*arcsin(c*x)+72*arcs in(c*x)^2-19)*cos(5*arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-5/55296*(-d*(c^2*x...
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2 *x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)
Output:
Timed out
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxim a")
Output:
1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d ) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a^2 + sqrt(d)*integrate(((b^2*c^4*d^2*x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1) )^2 + 2*(a*b*c^4*d^2*x^6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arctan2(c*x, s qrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Time = 1.70 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.35 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac" )
Output:
1/8*sqrt(-c^2*d*x^2 + d)*a^2*c^4*d^2*x^7 - 17/48*sqrt(-c^2*d*x^2 + d)*a^2* c^2*d^2*x^5 + 59/192*sqrt(-c^2*d*x^2 + d)*a^2*d^2*x^3 - 5/128*sqrt(-c^2*d* x^2 + d)*a^2*d^2*x/c^2 - 5/128*a^2*d^3*log(abs(-c*sqrt(-d)*x + sqrt(c^2*x^ 2 - 1)*sqrt(-d)))/(c^3*sqrt(-d)) + 1/110592*(13824*(c^2*x^2 - 1)^3*sqrt(-c ^2*x^2 + 1)*b^2*d^(5/2)*x*arcsin(c*x)^2 + 27648*(c^2*x^2 - 1)^3*sqrt(-c^2* x^2 + 1)*a*b*d^(5/2)*x*arcsin(c*x) + 2304*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^(5/2)*x*arcsin(c*x)^2 - 432*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^ 2*d^(5/2)*x + 4608*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^(5/2)*x*arcsin (c*x) + 2880*(-c^2*x^2 + 1)^(3/2)*b^2*d^(5/2)*x*arcsin(c*x)^2 - 3456*(c^2* x^2 - 1)^4*b^2*d^(5/2)*arcsin(c*x)/c + 376*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^(5/2)*x + 5760*(-c^2*x^2 + 1)^(3/2)*a*b*d^(5/2)*x*arcsin(c*x) + 4320*sqrt(-c^2*x^2 + 1)*b^2*d^(5/2)*x*arcsin(c*x)^2 - 3456*(c^2*x^2 - 1)^4 *a*b*d^(5/2)/c - 768*(c^2*x^2 - 1)^3*b^2*d^(5/2)*arcsin(c*x)/c + 110*(-c^2 *x^2 + 1)^(3/2)*b^2*d^(5/2)*x + 8640*sqrt(-c^2*x^2 + 1)*a*b*d^(5/2)*x*arcs in(c*x) - 768*(c^2*x^2 - 1)^3*a*b*d^(5/2)/c + 1440*(c^2*x^2 - 1)^2*b^2*d^( 5/2)*arcsin(c*x)/c + 1440*b^2*d^(5/2)*arcsin(c*x)^3/c - 1995*sqrt(-c^2*x^2 + 1)*b^2*d^(5/2)*x + 1440*(c^2*x^2 - 1)^2*a*b*d^(5/2)/c - 4320*(c^2*x^2 - 1)*b^2*d^(5/2)*arcsin(c*x)/c + 4320*a*b*d^(5/2)*arcsin(c*x)^2/c - 4320*(c ^2*x^2 - 1)*a*b*d^(5/2)/c - 1995*b^2*d^(5/2)*arcsin(c*x)/c - 1995*a*b*d^(5 /2)/c)/c^2
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a^{2}+48 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{7} x^{7}-136 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+118 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x +768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{6}d x \right ) a b \,c^{7}-1536 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}-768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{384 c^{3}} \] Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))^2,x)
Output:
(sqrt(d)*d**2*(15*asin(c*x)*a**2 + 48*sqrt( - c**2*x**2 + 1)*a**2*c**7*x** 7 - 136*sqrt( - c**2*x**2 + 1)*a**2*c**5*x**5 + 118*sqrt( - c**2*x**2 + 1) *a**2*c**3*x**3 - 15*sqrt( - c**2*x**2 + 1)*a**2*c*x + 768*int(sqrt( - c** 2*x**2 + 1)*asin(c*x)*x**6,x)*a*b*c**7 - 1536*int(sqrt( - c**2*x**2 + 1)*a sin(c*x)*x**4,x)*a*b*c**5 + 768*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**2, x)*a*b*c**3 + 384*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**6,x)*b**2*c** 7 - 768*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**4,x)*b**2*c**5 + 384*in t(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**2,x)*b**2*c**3))/(384*c**3)