Integrand size = 31, antiderivative size = 1281 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {1-c^2 x^2}}+\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {1-c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{24} d^2 f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{6} d^2 f^3 x \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {3 d^2 f^2 g \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c^2}-\frac {d^2 g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c^4}+\frac {d^2 g^3 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \]
5/96*b*c^3*d^2*f^3*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/21*b*c*d^ 2*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+19/441*b*c^3*d^2*g^3*x^7 *(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/81*b*c^5*d^2*g^3*x^9*(-c^2*d*x^ 2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/32*d^2*f^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2 +d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-15/128*d^2*f*g^2*x*(a+b*arcsin(c*x))*(-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))*(-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))*(-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))*(-c^2*d*x^2 +d)^(1/2)/c^2+2/63*b*d^2*g^3*x*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2) -25/96*b*c*d^2*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/189*b*d^2 *g^3*x^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+5/16*d^2*f^3*x*(a+b*arc sin(c*x))*(-c^2*d*x^2+d)^(1/2)+3/7*b*d^2*f^2*g*x*(-c^2*d*x^2+d)^(1/2)/c/(- c^2*x^2+1)^(1/2)+15/256*b*d^2*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1 )^(1/2)+1/36*b*d^2*f^3*(-c^2*x^2+1)^(5/2)*(-c^2*d*x^2+d)^(1/2)/c+15/64*d^2 *f*g^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+5/24*d^2*f^3*x*(-c^2*x^2 +1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+1/6*d^2*f^3*x*(-c^2*x^2+1)^2*(a +b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-1/7*d^2*g^3*(-c^2*x^2+1)^3*(a+b*arcsi n(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+1/9*d^2*g^3*(-c^2*x^2+1)^4*(a+b*arcsin(c* x))*(-c^2*d*x^2+d)^(1/2)/c^4-3/7*b*c*d^2*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(- c^2*x^2+1)^(1/2)-59/256*b*c*d^2*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^...
Time = 0.75 (sec) , antiderivative size = 587, normalized size of antiderivative = 0.46 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (99225 a^2 \left (8 c^3 f^3+3 c f g^2\right )+630 a b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )+b^2 c x \left (161280 g^3+105 c^2 g \left (20736 f^2+2835 f g x+256 g^2 x^2\right )-945 c^4 x \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )+72 c^6 x^3 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )-20 c^8 x^5 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )+630 b \left (315 a \left (8 c^3 f^3+3 c f g^2\right )+b \sqrt {1-c^2 x^2} \left (-256 g^3-c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )-8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )\right ) \arcsin (c x)+99225 b^2 c f \left (8 c^2 f^2+3 g^2\right ) \arcsin (c x)^2\right )}{5080320 b c^4 \sqrt {1-c^2 x^2}} \]
(d^2*Sqrt[d - c^2*d*x^2]*(99225*a^2*(8*c^3*f^3 + 3*c*f*g^2) + 630*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 + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(5 46*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*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*(7056*f^3 + 15552*f^2 *g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)) + 630*b*(315*a*(8*c^3*f^3 + 3*c*f* g^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)))*ArcSin[c*x] + 99225*b^2*c*f*(8*c^2*f^2 + 3*g^2)*ArcSin[c*x]^2))/(5080320*b*c^4*Sqrt[1 - c^2*x^2])
Time = 1.34 (sec) , antiderivative size = 630, normalized size of antiderivative = 0.49, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{5/2} (f+g x)^3 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) f^3+3 g x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x)) f+g^3 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {15 f g^2 (a+b \arcsin (c x))^2}{256 b c^3}+\frac {1}{6} f^3 x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{24} f^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {5}{16} f^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {3 f^2 g \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^2}-\frac {15 f g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {3}{8} f g^2 x^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {5}{16} f g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {15}{64} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {g^3 \left (1-c^2 x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^4}-\frac {g^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^4}+\frac {5 f^3 (a+b \arcsin (c x))^2}{32 b c}-\frac {3}{49} b c^5 f^2 g x^7-\frac {3}{64} b c^5 f g^2 x^8-\frac {1}{81} b c^5 g^3 x^9+\frac {5}{96} b c^3 f^3 x^4+\frac {9}{35} b c^3 f^2 g x^5+\frac {17}{96} b c^3 f g^2 x^6+\frac {19}{441} b c^3 g^3 x^7+\frac {2 b g^3 x}{63 c^3}+\frac {b f^3 \left (1-c^2 x^2\right )^3}{36 c}-\frac {25}{96} b c f^3 x^2-\frac {3}{7} b c f^2 g x^3+\frac {3 b f^2 g x}{7 c}-\frac {59}{256} b c f g^2 x^4+\frac {15 b f g^2 x^2}{256 c}-\frac {1}{21} b c g^3 x^5+\frac {b g^3 x^3}{189 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((3*b*f^2*g*x)/(7*c) + (2*b*g^3*x)/(63*c^3) - (25 *b*c*f^3*x^2)/96 + (15*b*f*g^2*x^2)/(256*c) - (3*b*c*f^2*g*x^3)/7 + (b*g^3 *x^3)/(189*c) + (5*b*c^3*f^3*x^4)/96 - (59*b*c*f*g^2*x^4)/256 + (9*b*c^3*f ^2*g*x^5)/35 - (b*c*g^3*x^5)/21 + (17*b*c^3*f*g^2*x^6)/96 - (3*b*c^5*f^2*g *x^7)/49 + (19*b*c^3*g^3*x^7)/441 - (3*b*c^5*f*g^2*x^8)/64 - (b*c^5*g^3*x^ 9)/81 + (b*f^3*(1 - c^2*x^2)^3)/(36*c) + (5*f^3*x*Sqrt[1 - c^2*x^2]*(a + b *ArcSin[c*x]))/16 - (15*f*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(12 8*c^2) + (15*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/64 + (5*f^3* x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/24 + (5*f*g^2*x^3*(1 - c^2*x^2) ^(3/2)*(a + b*ArcSin[c*x]))/16 + (f^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[ c*x]))/6 + (3*f*g^2*x^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 - (3*f^ 2*g*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^2) - (g^3*(1 - c^2*x^2)^ (7/2)*(a + b*ArcSin[c*x]))/(7*c^4) + (g^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSi n[c*x]))/(9*c^4) + (5*f^3*(a + b*ArcSin[c*x])^2)/(32*b*c) + (15*f*g^2*(a + b*ArcSin[c*x])^2)/(256*b*c^3)))/Sqrt[1 - c^2*x^2]
3.1.40.3.1 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 = 0.77 (sec) , antiderivative size = 2903, normalized size of antiderivative = 2.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2903\) |
| parts | \(\text {Expression too large to display}\) | \(2903\) |
a*(f^3*(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^3*(-1/9*x^2*(-c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4 *(-c^2*d*x^2+d)^(7/2))+3*f*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))))))-3/7*f^2*g*(-c^2*d*x^2+d)^(7/2)/c^2/d)+b*(-5/256*(-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)^2*f*(8*c^2*f^ 2+3*g^2)*d^2+1/41472*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8-256 *I*(-c^2*x^2+1)^(1/2)*x^9*c^9+688*c^6*x^6+576*I*(-c^2*x^2+1)^(1/2)*x^7*c^7 -280*c^4*x^4-432*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+41*c^2*x^2+120*I*(-c^2*x^2+1 )^(1/2)*x^3*c^3-9*I*(-c^2*x^2+1)^(1/2)*x*c-1)*g^3*(I+9*arcsin(c*x))*d^2/c^ 4/(c^2*x^2-1)+3/25088*(-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)*g*( 4*I*c^2*f^2+28*arcsin(c*x)*c^2*f^2-I*g^2-7*arcsin(c*x)*g^2)*d^2/c^4/(c^2*x ^2-1)-1/9216*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*f *(58*I*c^2*f^2+192*arcsin(c*x)*c^2*f^2-39*I*g^2-36*arcsin(c*x)*g^2)*cos(5* arcsin(c*x))*d^2/c^3/(c^2*x^2-1)-3/640*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6- 28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)...
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
integral((a*c^4*d^2*g^3*x^7 + 3*a*c^4*d^2*f*g^2*x^6 + 3*a*d^2*f^2*g*x + a* d^2*f^3 + (3*a*c^4*d^2*f^2*g - 2*a*c^2*d^2*g^3)*x^5 + (a*c^4*d^2*f^3 - 6*a *c^2*d^2*f*g^2)*x^4 - (6*a*c^2*d^2*f^2*g - a*d^2*g^3)*x^3 - (2*a*c^2*d^2*f ^3 - 3*a*d^2*f*g^2)*x^2 + (b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3*b *d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b* c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d ), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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*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)*ar csin(c*x)/c^3)*a*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*g^3 - 3/7*(-c^2*d*x^2 + d)^(7/2)*a*f^2*g/( c^2*d) + sqrt(d)*integrate((b*c^4*d^2*g^3*x^7 + 3*b*c^4*d^2*f*g^2*x^6 + 3* b*d^2*f^2*g*x + b*d^2*f^3 + (3*b*c^4*d^2*f^2*g - 2*b*c^2*d^2*g^3)*x^5 + (b *c^4*d^2*f^3 - 6*b*c^2*d^2*f*g^2)*x^4 - (6*b*c^2*d^2*f^2*g - b*d^2*g^3)*x^ 3 - (2*b*c^2*d^2*f^3 - 3*b*d^2*f*g^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*ar ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]