Integrand size = 33, antiderivative size = 1533 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {64 b^2 d^2 f g \sqrt {d-c^2 d x^2}}{245 c^2}-\frac {245 b^2 d^2 f^2 x \sqrt {d-c^2 d x^2}}{1152}-\frac {359 b^2 d^2 g^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 g^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 g^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 g^2 x^7 \sqrt {d-c^2 d x^2}+\frac {32 b^2 d^2 f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}-\frac {65 b^2 d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{1728}+\frac {24 b^2 d^2 f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}-\frac {1}{108} b^2 d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}+\frac {4 b^2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {115 b^2 d^2 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}+\frac {359 b^2 d^2 g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 f^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 g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c \sqrt {1-c^2 x^2}}-\frac {4 b c d^2 f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{384 \sqrt {1-c^2 x^2}}+\frac {12 b c^3 d^2 f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {4 b c^5 d^2 f g x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{32 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 f^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 f^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 f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {5 d^2 g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {5}{48} d^2 g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} d^2 f^2 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{8} d^2 g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 d^2 f g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}}+\frac {5 d^2 g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{384 b c^3 \sqrt {1-c^2 x^2}} \]
32/735*b^2*d^2*f*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+24/1225*b^2*d^2*f *g*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2+4/7*b*d^2*f*g*x*(a+b*arcsin(c*x ))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-4/7*b*c*d^2*f*g*x^3*(a+b*arcs in(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+12/35*b*c^3*d^2*f*g*x^5*( a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-4/49*b*c^5*d^2*f* g*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+4/343*b^2* d^2*f*g*(-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+5/48*b*d^2*f^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*f^2*(-c^2*x^2+ 1)^(5/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c-2/7*d^2*f*g*(-c^2*x^2+1) ^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+115/1152*b^2*d^2*f^2*arcsi n(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+359/36864*b^2*d^2*g^2*arc sin(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+5/48*d^2*f^2*(a+b*arc sin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+5/384*d^2*g^2*(a+b *arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2 *f^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/128*b *d^2*g^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-5 9/384*b*c*d^2*g^2*x^4*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^ (1/2)+17/144*b*c^3*d^2*g^2*x^6*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^ 2*x^2+1)^(1/2)-1/32*b*c^5*d^2*g^2*x^8*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/ 2)/(-c^2*x^2+1)^(1/2)+64/245*b^2*d^2*f*g*(-c^2*d*x^2+d)^(1/2)/c^2-359/3...
Time = 0.94 (sec) , antiderivative size = 742, normalized size of antiderivative = 0.48 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (12348000 a^3 \left (8 c^2 f^2+g^2\right )-3360 a b^2 c^2 x \left (1960 c^2 f^2 x \left (99-39 c^2 x^2+8 c^4 x^4\right )+4608 f g \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+245 g^2 x \left (-45+177 c^2 x^2-136 c^4 x^4+36 c^6 x^6\right )\right )+352800 a^2 b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )-b^3 c \sqrt {1-c^2 x^2} \left (274400 c^2 f^2 x \left (897-194 c^2 x^2+32 c^4 x^4\right )+147456 f g \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+8575 g^2 x \left (1077+2158 c^2 x^2-1672 c^4 x^4+432 c^6 x^6\right )\right )+105 b \left (352800 a^2 \left (8 c^2 f^2+g^2\right )+b^2 \left (87955 g^2+1120 c^2 \left (2093 f^2+4608 f g x+315 g^2 x^2\right )-3360 c^4 x^2 \left (1848 f^2+1536 f g x+413 g^2 x^2\right )-640 c^8 x^6 \left (784 f^2+1152 f g x+441 g^2 x^2\right )+1792 c^6 x^4 \left (1365 f^2+1728 f g x+595 g^2 x^2\right )\right )+6720 a b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )\right ) \arcsin (c x)+352800 b^2 \left (105 a \left (8 c^2 f^2+g^2\right )+b c \sqrt {1-c^2 x^2} \left (768 f g \left (-1+c^2 x^2\right )^3+56 c^2 f^2 x \left (33-26 c^2 x^2+8 c^4 x^4\right )+7 g^2 x \left (-15+118 c^2 x^2-136 c^4 x^4+48 c^6 x^6\right )\right )\right ) \arcsin (c x)^2+12348000 b^3 \left (8 c^2 f^2+g^2\right ) \arcsin (c x)^3\right )}{948326400 b c^3 \sqrt {1-c^2 x^2}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(12348000*a^3*(8*c^2*f^2 + g^2) - 3360*a*b^2*c^2* x*(1960*c^2*f^2*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 4608*f*g*(-35 + 35*c^2*x ^2 - 21*c^4*x^4 + 5*c^6*x^6) + 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6)) + 352800*a^2*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^ 3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^ 2 - 136*c^4*x^4 + 48*c^6*x^6)) - b^3*c*Sqrt[1 - c^2*x^2]*(274400*c^2*f^2*x *(897 - 194*c^2*x^2 + 32*c^4*x^4) + 147456*f*g*(-2161 + 757*c^2*x^2 - 351* c^4*x^4 + 75*c^6*x^6) + 8575*g^2*x*(1077 + 2158*c^2*x^2 - 1672*c^4*x^4 + 4 32*c^6*x^6)) + 105*b*(352800*a^2*(8*c^2*f^2 + g^2) + b^2*(87955*g^2 + 1120 *c^2*(2093*f^2 + 4608*f*g*x + 315*g^2*x^2) - 3360*c^4*x^2*(1848*f^2 + 1536 *f*g*x + 413*g^2*x^2) - 640*c^8*x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728*f*g*x + 595*g^2*x^2)) + 6720*a*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^ 4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c *x] + 352800*b^2*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2]*(768*f*g *(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*( -15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x]^2 + 12348000*b ^3*(8*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(948326400*b*c^3*Sqrt[1 - c^2*x^2])
Time = 2.15 (sec) , antiderivative size = 922, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5276, 5262, 2009}
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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (f^2 (a+b \arcsin (c x))^2 \left (1-c^2 x^2\right )^{5/2}+g^2 x^2 (a+b \arcsin (c x))^2 \left (1-c^2 x^2\right )^{5/2}+2 f g x (a+b \arcsin (c x))^2 \left (1-c^2 x^2\right )^{5/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{32} b c^5 g^2 (a+b \arcsin (c x)) x^8-\frac {4}{49} b c^5 f g (a+b \arcsin (c x)) x^7-\frac {1}{256} b^2 c^4 g^2 \sqrt {1-c^2 x^2} x^7+\frac {17}{144} b c^3 g^2 (a+b \arcsin (c x)) x^6+\frac {12}{35} b c^3 f g (a+b \arcsin (c x)) x^5+\frac {209 b^2 c^2 g^2 \sqrt {1-c^2 x^2} x^5}{13824}-\frac {59}{384} b c g^2 (a+b \arcsin (c x)) x^4+\frac {1}{8} g^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 x^3+\frac {5}{48} g^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^3+\frac {5}{64} g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^3-\frac {4}{7} b c f g (a+b \arcsin (c x)) x^3-\frac {1079 b^2 g^2 \sqrt {1-c^2 x^2} x^3}{55296}-\frac {5}{16} b c f^2 (a+b \arcsin (c x)) x^2+\frac {5 b g^2 (a+b \arcsin (c x)) x^2}{128 c}-\frac {1}{108} b^2 f^2 \left (1-c^2 x^2\right )^{5/2} x+\frac {1}{6} f^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 x+\frac {5}{24} f^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x+\frac {5}{16} f^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x-\frac {5 g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x}{128 c^2}-\frac {65 b^2 f^2 \left (1-c^2 x^2\right )^{3/2} x}{1728}+\frac {4 b f g (a+b \arcsin (c x)) x}{7 c}-\frac {245 b^2 f^2 \sqrt {1-c^2 x^2} x}{1152}-\frac {359 b^2 g^2 \sqrt {1-c^2 x^2} x}{36864 c^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right )^{7/2}}{343 c^2}+\frac {5 f^2 (a+b \arcsin (c x))^3}{48 b c}+\frac {5 g^2 (a+b \arcsin (c x))^3}{384 b c^3}+\frac {24 b^2 f g \left (1-c^2 x^2\right )^{5/2}}{1225 c^2}-\frac {2 f g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2}+\frac {32 b^2 f g \left (1-c^2 x^2\right )^{3/2}}{735 c^2}+\frac {115 b^2 f^2 \arcsin (c x)}{1152 c}+\frac {359 b^2 g^2 \arcsin (c x)}{36864 c^3}+\frac {b f^2 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{18 c}+\frac {5 b f^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{48 c}+\frac {64 b^2 f g \sqrt {1-c^2 x^2}}{245 c^2}\right )}{\sqrt {1-c^2 x^2}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((64*b^2*f*g*Sqrt[1 - c^2*x^2])/(245*c^2) - (245* b^2*f^2*x*Sqrt[1 - c^2*x^2])/1152 - (359*b^2*g^2*x*Sqrt[1 - c^2*x^2])/(368 64*c^2) - (1079*b^2*g^2*x^3*Sqrt[1 - c^2*x^2])/55296 + (209*b^2*c^2*g^2*x^ 5*Sqrt[1 - c^2*x^2])/13824 - (b^2*c^4*g^2*x^7*Sqrt[1 - c^2*x^2])/256 + (32 *b^2*f*g*(1 - c^2*x^2)^(3/2))/(735*c^2) - (65*b^2*f^2*x*(1 - c^2*x^2)^(3/2 ))/1728 + (24*b^2*f*g*(1 - c^2*x^2)^(5/2))/(1225*c^2) - (b^2*f^2*x*(1 - c^ 2*x^2)^(5/2))/108 + (4*b^2*f*g*(1 - c^2*x^2)^(7/2))/(343*c^2) + (115*b^2*f ^2*ArcSin[c*x])/(1152*c) + (359*b^2*g^2*ArcSin[c*x])/(36864*c^3) + (4*b*f* g*x*(a + b*ArcSin[c*x]))/(7*c) - (5*b*c*f^2*x^2*(a + b*ArcSin[c*x]))/16 + (5*b*g^2*x^2*(a + b*ArcSin[c*x]))/(128*c) - (4*b*c*f*g*x^3*(a + b*ArcSin[c *x]))/7 - (59*b*c*g^2*x^4*(a + b*ArcSin[c*x]))/384 + (12*b*c^3*f*g*x^5*(a + b*ArcSin[c*x]))/35 + (17*b*c^3*g^2*x^6*(a + b*ArcSin[c*x]))/144 - (4*b*c ^5*f*g*x^7*(a + b*ArcSin[c*x]))/49 - (b*c^5*g^2*x^8*(a + b*ArcSin[c*x]))/3 2 + (5*b*f^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(48*c) + (b*f^2*(1 - c^2 *x^2)^3*(a + b*ArcSin[c*x]))/(18*c) + (5*f^2*x*Sqrt[1 - c^2*x^2]*(a + b*Ar cSin[c*x])^2)/16 - (5*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(128* c^2) + (5*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*f^2*x*( 1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/24 + (5*g^2*x^3*(1 - c^2*x^2)^(3 /2)*(a + b*ArcSin[c*x])^2)/48 + (f^2*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c *x])^2)/6 + (g^2*x^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/8 - (2*...
3.1.67.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & & EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ [n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 4170, normalized size of antiderivative = 2.72
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4170\) |
| parts | \(\text {Expression too large to display}\) | \(4170\) |
a^2*(f^2*(1/6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4 *d*(1/2*x*(-c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/ (-c^2*d*x^2+d)^(1/2)))))+g^2*(-1/8*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/8/c^2*(1 /6*x*(-c^2*d*x^2+d)^(5/2)+5/6*d*(1/4*x*(-c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*( -c^2*d*x^2+d)^(1/2)+1/2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2 +d)^(1/2))))))-2/7*f*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+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*(8*c^2*f^2+g^ 2)*d^2+1/21952*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7 *(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^ 2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*f*g*(14*I* arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1)-3/274400*(-d*(c^2*x^2- 1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f*g*(630*I*arcsin(c*x)+1225 *arcsin(c*x)^2-106)*sin(6*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)+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)*c^2*f^2+18*arcsin(c*x)^2*c^2*f^ 2-6*I*arcsin(c*x)*g^2-18*arcsin(c*x)^2*g^2-c^2*f^2+g^2)*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+32*I*(-c^2*x^2+1)^(1/2)*c^2*x^2-88*c^3*x^3-I*(-c^...
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral((a^2*c^4*d^2*g^2*x^6 + 2*a^2*c^4*d^2*f*g*x^5 - 4*a^2*c^2*d^2*f*g* x^3 + 2*a^2*d^2*f*g*x + a^2*d^2*f^2 + (a^2*c^4*d^2*f^2 - 2*a^2*c^2*d^2*g^2 )*x^4 - (2*a^2*c^2*d^2*f^2 - a^2*d^2*g^2)*x^2 + (b^2*c^4*d^2*g^2*x^6 + 2*b ^2*c^4*d^2*f*g*x^5 - 4*b^2*c^2*d^2*f*g*x^3 + 2*b^2*d^2*f*g*x + b^2*d^2*f^2 + (b^2*c^4*d^2*f^2 - 2*b^2*c^2*d^2*g^2)*x^4 - (2*b^2*c^2*d^2*f^2 - b^2*d^ 2*g^2)*x^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*g^2*x^6 + 2*a*b*c^4*d^2*f*g*x^5 - 4*a*b*c^2*d^2*f*g*x^3 + 2*a*b*d^2*f*g*x + a*b*d^2*f^2 + (a*b*c^4*d^2*f^ 2 - 2*a*b*c^2*d^2*g^2)*x^4 - (2*a*b*c^2*d^2*f^2 - a*b*d^2*g^2)*x^2)*arcsin (c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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*f^2 + 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*g^2 - 2/7*(-c^2*d*x^2 + d)^(7/2)*a^2*f*g/(c^2*d) + sq rt(d)*integrate(((b^2*c^4*d^2*g^2*x^6 + 2*b^2*c^4*d^2*f*g*x^5 - 4*b^2*c^2* d^2*f*g*x^3 + 2*b^2*d^2*f*g*x + b^2*d^2*f^2 + (b^2*c^4*d^2*f^2 - 2*b^2*c^2 *d^2*g^2)*x^4 - (2*b^2*c^2*d^2*f^2 - b^2*d^2*g^2)*x^2)*arctan2(c*x, sqrt(c *x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*g^2*x^6 + 2*a*b*c^4*d^2*f*g*x^5 - 4*a*b*c^2*d^2*f*g*x^3 + 2*a*b*d^2*f*g*x + a*b*d^2*f^2 + (a*b*c^4*d^2*f^ 2 - 2*a*b*c^2*d^2*g^2)*x^4 - (2*a*b*c^2*d^2*f^2 - a*b*d^2*g^2)*x^2)*arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]