Integrand size = 33, antiderivative size = 2290 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \]
1/18*b*d^2*f^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c -15/128*d^2*f*g^2*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+5/16*d^2* f*g^2*x^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+3/8*d^2*f* g^2*x^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-3/7*d^2*f^ 2*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^3*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+5/48*d^2* f^3*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-359/12 288*b^2*d^2*f*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+209/4608*b^2*c^2*d^2*f*g^2*x^ 5*(-c^2*d*x^2+d)^(1/2)-3/256*b^2*c^4*d^2*f*g^2*x^7*(-c^2*d*x^2+d)^(1/2)+16 /245*b^2*d^2*f^2*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+36/1225*b^2*d^2*f ^2*g*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2+6/343*b^2*d^2*f^2*g*(-c^2*x^2 +1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+5/48*b*d^2*f^3*(-c^2*x^2+1)^(3/2)*(a+b*arcs in(c*x))*(-c^2*d*x^2+d)^(1/2)/c-59/128*b*c*d^2*f*g^2*x^4*(a+b*arcsin(c*x)) *(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+18/35*b*c^3*d^2*f^2*g*x^5*(a+b*ar csin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+17/48*b*c^3*d^2*f*g^2*x ^6*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-6/49*b*c^5*d^ 2*f^2*g*x^7*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/32 *b*c^5*d^2*f*g^2*x^8*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^( 1/2)+6/7*b*d^2*f^2*g*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+ 1)^(1/2)+15/128*b*d^2*f*g^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/...
Time = 1.28 (sec) , antiderivative size = 1114, normalized size of antiderivative = 0.49 \[ \int (f+g x)^3 \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 (333396000 a^3 \left (8 c^3 f^3+3 c f g^2\right )+3175200 a^2 b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )-10080 a b^2 c x \left (-161280 g^3-105 c^2 g \left (20736 f^2+2835 f g x+256 g^2 x^2\right )+945 c^4 x \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )-72 c^6 x^3 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )+20 c^8 x^5 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )-b^3 \sqrt {1-c^2 x^2} \left (-1257472000 g^3+c^2 g \left (-12905422848 f^2+748057275 f g x+184115200 g^2 x^2\right )+400 c^8 x^5 \left (592704 f^3+1119744 f^2 g x+750141 f g^2 x^2+175616 g^3 x^3\right )-8 c^6 x^3 \left (179663400 f^3+262020096 f^2 g x+145166175 f g^2 x^2+29363200 g^3 x^3\right )+6 c^4 x \left (1107615600 f^3+753463296 f^2 g x+249815475 f g^2 x^2+34304000 g^3 x^3\right )\right )+315 b \left (3175200 a^2 \left (8 c^3 f^3+3 c f g^2\right )+20160 a b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )+b^2 c \left (315 g^2 (7539 f+16384 g x)-30240 c^4 x^2 \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )+3360 c^2 \left (6279 f^3+20736 f^2 g x+2835 f g^2 x^2+256 g^3 x^3\right )+2304 c^6 x^4 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )-640 c^8 x^6 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )\right ) \arcsin (c x)+3175200 b^2 \left (315 a \left (8 c^3 f^3+3 c f g^2\right )+b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )\right ) \arcsin (c x)^2+333396000 b^3 c f \left (8 c^2 f^2+3 g^2\right ) \arcsin (c x)^3\right )}{25604812800 b c^4 \sqrt {1-c^2 x^2}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(333396000*a^3*(8*c^3*f^3 + 3*c*f*g^2) + 3175200* a^2*b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2* x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8* c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x* (924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)) - 10080*a*b^2*c*x *(-161280*g^3 - 105*c^2*g*(20736*f^2 + 2835*f*g*x + 256*g^2*x^2) + 945*c^4 *x*(1848*f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) - 72*c^6*x^3*( 9555*f^3 + 18144*f^2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) + 20*c^8*x^5*(7 056*f^3 + 15552*f^2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)) - b^3*Sqrt[1 - c^2*x^2]*(-1257472000*g^3 + c^2*g*(-12905422848*f^2 + 748057275*f*g*x + 18 4115200*g^2*x^2) + 400*c^8*x^5*(592704*f^3 + 1119744*f^2*g*x + 750141*f*g^ 2*x^2 + 175616*g^3*x^3) - 8*c^6*x^3*(179663400*f^3 + 262020096*f^2*g*x + 1 45166175*f*g^2*x^2 + 29363200*g^3*x^3) + 6*c^4*x*(1107615600*f^3 + 7534632 96*f^2*g*x + 249815475*f*g^2*x^2 + 34304000*g^3*x^3)) + 315*b*(3175200*a^2 *(8*c^3*f^3 + 3*c*f*g^2) + 20160*a*b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*( 3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 1 89*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^ 2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)) + b^2*c*(315*g^2*(7539*f + 16384*g*x) - 30240*c^4*x^2*(1848* f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) + 3360*c^2*(6279*f^3...
Time = 3.24 (sec) , antiderivative size = 1379, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 f^3+3 g x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 f+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {2}{81} b c^5 g^3 (a+b \arcsin (c x)) x^9-\frac {3}{32} b c^5 f g^2 (a+b \arcsin (c x)) x^8+\frac {38}{441} b c^3 g^3 (a+b \arcsin (c x)) x^7-\frac {6}{49} b c^5 f^2 g (a+b \arcsin (c x)) x^7-\frac {3}{256} b^2 c^4 f g^2 \sqrt {1-c^2 x^2} x^7+\frac {17}{48} b c^3 f g^2 (a+b \arcsin (c x)) x^6-\frac {2}{21} b c g^3 (a+b \arcsin (c x)) x^5+\frac {18}{35} b c^3 f^2 g (a+b \arcsin (c x)) x^5+\frac {209 b^2 c^2 f g^2 \sqrt {1-c^2 x^2} x^5}{4608}+\frac {1}{9} g^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 x^4+\frac {5}{63} g^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^4+\frac {1}{21} g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^4-\frac {59}{128} b c f g^2 (a+b \arcsin (c x)) x^4+\frac {3}{8} f g^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 x^3+\frac {5}{16} f g^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x^3+\frac {15}{64} f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^3+\frac {2 b g^3 (a+b \arcsin (c x)) x^3}{189 c}-\frac {6}{7} b c f^2 g (a+b \arcsin (c x)) x^3-\frac {1079 b^2 f g^2 \sqrt {1-c^2 x^2} x^3}{18432}-\frac {g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x^2}{63 c^2}-\frac {5}{16} b c f^3 (a+b \arcsin (c x)) x^2+\frac {15 b f g^2 (a+b \arcsin (c x)) x^2}{128 c}+\frac {4 a b g^3 x}{63 c^3}-\frac {1}{108} b^2 f^3 \left (1-c^2 x^2\right )^{5/2} x+\frac {1}{6} f^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2 x+\frac {5}{24} f^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2 x+\frac {5}{16} f^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x-\frac {15 f g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 x}{128 c^2}-\frac {65 b^2 f^3 \left (1-c^2 x^2\right )^{3/2} x}{1728}+\frac {4 b^2 g^3 \arcsin (c x) x}{63 c^3}+\frac {6 b f^2 g (a+b \arcsin (c x)) x}{7 c}-\frac {245 b^2 f^3 \sqrt {1-c^2 x^2} x}{1152}-\frac {359 b^2 f g^2 \sqrt {1-c^2 x^2} x}{12288 c^2}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^{9/2}}{729 c^4}+\frac {50 b^2 g^3 \left (1-c^2 x^2\right )^{7/2}}{27783 c^4}+\frac {6 b^2 f^2 g \left (1-c^2 x^2\right )^{7/2}}{343 c^2}+\frac {5 f^3 (a+b \arcsin (c x))^3}{48 b c}+\frac {5 f g^2 (a+b \arcsin (c x))^3}{128 b c^3}+\frac {4 b^2 g^3 \left (1-c^2 x^2\right )^{5/2}}{1323 c^4}+\frac {36 b^2 f^2 g \left (1-c^2 x^2\right )^{5/2}}{1225 c^2}-\frac {3 f^2 g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2}-\frac {2 g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{63 c^4}+\frac {80 b^2 g^3 \left (1-c^2 x^2\right )^{3/2}}{11907 c^4}+\frac {16 b^2 f^2 g \left (1-c^2 x^2\right )^{3/2}}{245 c^2}+\frac {115 b^2 f^3 \arcsin (c x)}{1152 c}+\frac {359 b^2 f g^2 \arcsin (c x)}{12288 c^3}+\frac {b f^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{18 c}+\frac {5 b f^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{48 c}+\frac {160 b^2 g^3 \sqrt {1-c^2 x^2}}{3969 c^4}+\frac {96 b^2 f^2 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]*((4*a*b*g^3*x)/(63*c^3) + (96*b^2*f^2*g*Sqrt[1 - c^2*x^2])/(245*c^2) + (160*b^2*g^3*Sqrt[1 - c^2*x^2])/(3969*c^4) - (245*b^ 2*f^3*x*Sqrt[1 - c^2*x^2])/1152 - (359*b^2*f*g^2*x*Sqrt[1 - c^2*x^2])/(122 88*c^2) - (1079*b^2*f*g^2*x^3*Sqrt[1 - c^2*x^2])/18432 + (209*b^2*c^2*f*g^ 2*x^5*Sqrt[1 - c^2*x^2])/4608 - (3*b^2*c^4*f*g^2*x^7*Sqrt[1 - c^2*x^2])/25 6 + (16*b^2*f^2*g*(1 - c^2*x^2)^(3/2))/(245*c^2) + (80*b^2*g^3*(1 - c^2*x^ 2)^(3/2))/(11907*c^4) - (65*b^2*f^3*x*(1 - c^2*x^2)^(3/2))/1728 + (36*b^2* f^2*g*(1 - c^2*x^2)^(5/2))/(1225*c^2) + (4*b^2*g^3*(1 - c^2*x^2)^(5/2))/(1 323*c^4) - (b^2*f^3*x*(1 - c^2*x^2)^(5/2))/108 + (6*b^2*f^2*g*(1 - c^2*x^2 )^(7/2))/(343*c^2) + (50*b^2*g^3*(1 - c^2*x^2)^(7/2))/(27783*c^4) - (2*b^2 *g^3*(1 - c^2*x^2)^(9/2))/(729*c^4) + (115*b^2*f^3*ArcSin[c*x])/(1152*c) + (359*b^2*f*g^2*ArcSin[c*x])/(12288*c^3) + (4*b^2*g^3*x*ArcSin[c*x])/(63*c ^3) + (6*b*f^2*g*x*(a + b*ArcSin[c*x]))/(7*c) - (5*b*c*f^3*x^2*(a + b*ArcS in[c*x]))/16 + (15*b*f*g^2*x^2*(a + b*ArcSin[c*x]))/(128*c) - (6*b*c*f^2*g *x^3*(a + b*ArcSin[c*x]))/7 + (2*b*g^3*x^3*(a + b*ArcSin[c*x]))/(189*c) - (59*b*c*f*g^2*x^4*(a + b*ArcSin[c*x]))/128 + (18*b*c^3*f^2*g*x^5*(a + b*Ar cSin[c*x]))/35 - (2*b*c*g^3*x^5*(a + b*ArcSin[c*x]))/21 + (17*b*c^3*f*g^2* x^6*(a + b*ArcSin[c*x]))/48 - (6*b*c^5*f^2*g*x^7*(a + b*ArcSin[c*x]))/49 + (38*b*c^3*g^3*x^7*(a + b*ArcSin[c*x]))/441 - (3*b*c^5*f*g^2*x^8*(a + b*Ar cSin[c*x]))/32 - (2*b*c^5*g^3*x^9*(a + b*ArcSin[c*x]))/81 + (5*b*f^3*(1...
3.1.66.3.1 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 = 1.22 (sec) , antiderivative size = 5977, normalized size of antiderivative = 2.61
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(5977\) |
| parts | \(\text {Expression too large to display}\) | \(5977\) |
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
integral((a^2*c^4*d^2*g^3*x^7 + 3*a^2*c^4*d^2*f*g^2*x^6 + 3*a^2*d^2*f^2*g* x + a^2*d^2*f^3 + (3*a^2*c^4*d^2*f^2*g - 2*a^2*c^2*d^2*g^3)*x^5 + (a^2*c^4 *d^2*f^3 - 6*a^2*c^2*d^2*f*g^2)*x^4 - (6*a^2*c^2*d^2*f^2*g - a^2*d^2*g^3)* x^3 - (2*a^2*c^2*d^2*f^3 - 3*a^2*d^2*f*g^2)*x^2 + (b^2*c^4*d^2*g^3*x^7 + 3 *b^2*c^4*d^2*f*g^2*x^6 + 3*b^2*d^2*f^2*g*x + b^2*d^2*f^3 + (3*b^2*c^4*d^2* f^2*g - 2*b^2*c^2*d^2*g^3)*x^5 + (b^2*c^4*d^2*f^3 - 6*b^2*c^2*d^2*f*g^2)*x ^4 - (6*b^2*c^2*d^2*f^2*g - b^2*d^2*g^3)*x^3 - (2*b^2*c^2*d^2*f^3 - 3*b^2* d^2*f*g^2)*x^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*g^3*x^7 + 3*a*b*c^4*d^2*f*g ^2*x^6 + 3*a*b*d^2*f^2*g*x + a*b*d^2*f^3 + (3*a*b*c^4*d^2*f^2*g - 2*a*b*c^ 2*d^2*g^3)*x^5 + (a*b*c^4*d^2*f^3 - 6*a*b*c^2*d^2*f*g^2)*x^4 - (6*a*b*c^2* d^2*f^2*g - a*b*d^2*g^3)*x^3 - (2*a*b*c^2*d^2*f^3 - 3*a*b*d^2*f*g^2)*x^2)* arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \]
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^3 + 1/128*(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*f*g^2 - 1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a^2*g^3 - 3/7*(-c^2*d*x^2 + d)^(7/2)*a^2 *f^2*g/(c^2*d) + sqrt(d)*integrate(((b^2*c^4*d^2*g^3*x^7 + 3*b^2*c^4*d^2*f *g^2*x^6 + 3*b^2*d^2*f^2*g*x + b^2*d^2*f^3 + (3*b^2*c^4*d^2*f^2*g - 2*b^2* c^2*d^2*g^3)*x^5 + (b^2*c^4*d^2*f^3 - 6*b^2*c^2*d^2*f*g^2)*x^4 - (6*b^2*c^ 2*d^2*f^2*g - b^2*d^2*g^3)*x^3 - (2*b^2*c^2*d^2*f^3 - 3*b^2*d^2*f*g^2)*x^2 )*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*g^3*x^7 + 3*a*b*c^4*d^2*f*g^2*x^6 + 3*a*b*d^2*f^2*g*x + a*b*d^2*f^3 + (3*a*b*c^4*d^2 *f^2*g - 2*a*b*c^2*d^2*g^3)*x^5 + (a*b*c^4*d^2*f^3 - 6*a*b*c^2*d^2*f*g^2)* x^4 - (6*a*b*c^2*d^2*f^2*g - a*b*d^2*g^3)*x^3 - (2*a*b*c^2*d^2*f^3 - 3*a*b *d^2*f*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1) *sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]