Integrand size = 33, antiderivative size = 1044 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 f g \left (d-c^2 d x^2\right )^{3/2}}{225 c^2}-\frac {1}{32} b^2 f^2 x \left (d-c^2 d x^2\right )^{3/2}+\frac {4 b^2 f g \left (d-c^2 d x^2\right )^{5/2}}{125 c^2 d}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{4} f^2 x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{6} g^2 x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 f g \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {d f^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \] Output:
-1/16*d*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^2+32/75*b^2*d*f*g *(-c^2*d*x^2+d)^(1/2)/c^2-7/1152*b^2*d*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+1/10 8*b^2*c^2*d*g^2*x^5*(-c^2*d*x^2+d)^(1/2)+4/125*b^2*f*g*(-c^2*d*x^2+d)^(5/2 )/c^2/d-2/5*f*g*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2/c^2/d+16/225*b^2* f*g*(-c^2*d*x^2+d)^(3/2)/c^2-3/8*b*c*d*f^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*a rcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/16*b*d*g^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b* arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*g^2*x^4*(-c^2*d*x^2+d)^(1/2)* (a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*g^2*x^6*(-c^2*d*x^2+d)^( 1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/8*b*d*f^2*(-c^2*x^2+1)^(3/2)*( -c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c+1/8*d*f^2*(-c^2*d*x^2+d)^(1/2)*(a+ b*arcsin(c*x))^3/b/c/(-c^2*x^2+1)^(1/2)+1/48*d*g^2*(-c^2*d*x^2+d)^(1/2)*(a +b*arcsin(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2)+9/64*b^2*d*f^2*(-c^2*d*x^2+d)^( 1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)+7/1152*b^2*d*g^2*(-c^2*d*x^2+d)^(1/2 )*arcsin(c*x)/c^3/(-c^2*x^2+1)^(1/2)+1/4*f^2*x*(-c^2*d*x^2+d)^(3/2)*(a+b*a rcsin(c*x))^2+1/6*g^2*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-1/32*b^ 2*f^2*x*(-c^2*d*x^2+d)^(3/2)-15/64*b^2*d*f^2*x*(-c^2*d*x^2+d)^(1/2)-43/172 8*b^2*d*g^2*x^3*(-c^2*d*x^2+d)^(1/2)+3/8*d*f^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b *arcsin(c*x))^2+1/8*d*g^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2-8/1 5*b*c*d*f*g*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+ 4/25*b*c^3*d*f*g*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1...
Time = 0.60 (sec) , antiderivative size = 616, normalized size of antiderivative = 0.59 \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (9000 a^3 \left (6 c^2 f^2+g^2\right )+120 a b^2 c^2 x \left (450 c^2 f^2 x \left (-5+c^2 x^2\right )+192 f g \left (15-10 c^2 x^2+3 c^4 x^4\right )+25 g^2 x \left (9-21 c^2 x^2+8 c^4 x^4\right )\right )-1800 a^2 b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )+b^3 c \sqrt {1-c^2 x^2} \left (6750 c^2 f^2 x \left (-17+2 c^2 x^2\right )+1536 f g \left (149-38 c^2 x^2+9 c^4 x^4\right )+125 g^2 x \left (-21-86 c^2 x^2+32 c^4 x^4\right )\right )+15 b \left (1800 a^2 \left (6 c^2 f^2+g^2\right )+b^2 \left (175 g^2+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )-120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )\right )-240 a b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)+1800 b^2 \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt {1-c^2 x^2} \left (96 f g \left (-1+c^2 x^2\right )^2+30 c^2 f^2 x \left (-5+2 c^2 x^2\right )+5 g^2 x \left (3-14 c^2 x^2+8 c^4 x^4\right )\right )\right ) \arcsin (c x)^2+9000 b^3 \left (6 c^2 f^2+g^2\right ) \arcsin (c x)^3\right )}{432000 b c^3 \sqrt {1-c^2 x^2}} \] Input:
Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(d*Sqrt[d - c^2*d*x^2]*(9000*a^3*(6*c^2*f^2 + g^2) + 120*a*b^2*c^2*x*(450* c^2*f^2*x*(-5 + c^2*x^2) + 192*f*g*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 25*g^2* x*(9 - 21*c^2*x^2 + 8*c^4*x^4)) - 1800*a^2*b*c*Sqrt[1 - c^2*x^2]*(96*f*g*( -1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)) + b^3*c*Sqrt[1 - c^2*x^2]*(6750*c^2*f^2*x*(-17 + 2*c^2*x^2) + 1536*f*g*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 125*g^2*x*(-21 - 86*c^2*x^2 + 32*c^4*x^4)) + 15*b*(1800*a^2*(6*c^2*f^2 + g^2) + b^2*(175*g^2 + 90*c^2*(8 5*f^2 + 256*f*g*x + 20*g^2*x^2) - 120*c^4*x^2*(150*f^2 + 128*f*g*x + 35*g^ 2*x^2) + 16*c^6*x^4*(225*f^2 + 288*f*g*x + 100*g^2*x^2)) - 240*a*b*c*Sqrt[ 1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5* g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 1800*b^2*(15*a*(6*c^2*f ^2 + g^2) - b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x* (-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x]^2 + 9000*b^3*(6*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(432000*b*c^3*Sqrt[1 - c^2*x^2] )
Time = 1.72 (sec) , antiderivative size = 699, normalized size of antiderivative = 0.67, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x)^2 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (f^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+2 f g x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {4}{25} b c^3 f g x^5 (a+b \arcsin (c x))+\frac {1}{18} b c^3 g^2 x^6 (a+b \arcsin (c x))+\frac {g^2 (a+b \arcsin (c x))^3}{48 b c^3}+\frac {1}{4} f^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{8} f^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {b f^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{8 c}-\frac {2 f g \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}-\frac {g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{6} g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{8} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {3}{8} b c f^2 x^2 (a+b \arcsin (c x))+\frac {f^2 (a+b \arcsin (c x))^3}{8 b c}-\frac {8}{15} b c f g x^3 (a+b \arcsin (c x))+\frac {4 b f g x (a+b \arcsin (c x))}{5 c}-\frac {7}{48} b c g^2 x^4 (a+b \arcsin (c x))+\frac {b g^2 x^2 (a+b \arcsin (c x))}{16 c}+\frac {7 b^2 g^2 \arcsin (c x)}{1152 c^3}+\frac {9 b^2 f^2 \arcsin (c x)}{64 c}-\frac {1}{32} b^2 f^2 x \left (1-c^2 x^2\right )^{3/2}-\frac {15}{64} b^2 f^2 x \sqrt {1-c^2 x^2}+\frac {4 b^2 f g \left (1-c^2 x^2\right )^{5/2}}{125 c^2}+\frac {16 b^2 f g \left (1-c^2 x^2\right )^{3/2}}{225 c^2}+\frac {32 b^2 f g \sqrt {1-c^2 x^2}}{75 c^2}-\frac {7 b^2 g^2 x \sqrt {1-c^2 x^2}}{1152 c^2}+\frac {1}{108} b^2 c^2 g^2 x^5 \sqrt {1-c^2 x^2}-\frac {43 b^2 g^2 x^3 \sqrt {1-c^2 x^2}}{1728}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]
Output:
(d*Sqrt[d - c^2*d*x^2]*((32*b^2*f*g*Sqrt[1 - c^2*x^2])/(75*c^2) - (15*b^2* f^2*x*Sqrt[1 - c^2*x^2])/64 - (7*b^2*g^2*x*Sqrt[1 - c^2*x^2])/(1152*c^2) - (43*b^2*g^2*x^3*Sqrt[1 - c^2*x^2])/1728 + (b^2*c^2*g^2*x^5*Sqrt[1 - c^2*x ^2])/108 + (16*b^2*f*g*(1 - c^2*x^2)^(3/2))/(225*c^2) - (b^2*f^2*x*(1 - c^ 2*x^2)^(3/2))/32 + (4*b^2*f*g*(1 - c^2*x^2)^(5/2))/(125*c^2) + (9*b^2*f^2* ArcSin[c*x])/(64*c) + (7*b^2*g^2*ArcSin[c*x])/(1152*c^3) + (4*b*f*g*x*(a + b*ArcSin[c*x]))/(5*c) - (3*b*c*f^2*x^2*(a + b*ArcSin[c*x]))/8 + (b*g^2*x^ 2*(a + b*ArcSin[c*x]))/(16*c) - (8*b*c*f*g*x^3*(a + b*ArcSin[c*x]))/15 - ( 7*b*c*g^2*x^4*(a + b*ArcSin[c*x]))/48 + (4*b*c^3*f*g*x^5*(a + b*ArcSin[c*x ]))/25 + (b*c^3*g^2*x^6*(a + b*ArcSin[c*x]))/18 + (b*f^2*(1 - c^2*x^2)^2*( a + b*ArcSin[c*x]))/(8*c) + (3*f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) ^2)/8 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (g^2*x^ 3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/8 + (f^2*x*(1 - c^2*x^2)^(3/2)* (a + b*ArcSin[c*x])^2)/4 + (g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x] )^2)/6 - (2*f*g*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*c^2) + (f^2* (a + b*ArcSin[c*x])^3)/(8*b*c) + (g^2*(a + b*ArcSin[c*x])^3)/(48*b*c^3)))/ Sqrt[1 - c^2*x^2]
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 = 1.11 (sec) , antiderivative size = 3032, normalized size of antiderivative = 2.90
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3032\) |
| parts | \(\text {Expression too large to display}\) | \(3032\) |
Input:
int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVER BOSE)
Output:
a^2*(f^2*(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*(-1/6* x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/6/c^2*(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)))))-2/5*f*g*(-c^2*d*x^2+d)^(5/2)/c^2/d)+b^2*(-1/48*(-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*(6*c^2*f^2 +g^2)*d-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)*g^2*(6*I*arcsin(c*x)+18*a rcsin(c*x)^2-1)*d/c^3/(c^2*x^2-1)-1/2000*(-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)^ (1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*f*g*(10*I*arcsin(c*x)+25*arcsi n(c*x)^2-2)*d/c^2/(c^2*x^2-1)-1/1024*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^ 2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*( -c^2*x^2+1)^(1/2)+4*c*x)*(8*I*arcsin(c*x)*c^2*f^2+16*arcsin(c*x)^2*c^2*f^2 -4*I*arcsin(c*x)*g^2-8*arcsin(c*x)^2*g^2-2*c^2*f^2+g^2)*d/c^3/(c^2*x^2-1)- 1/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*f*g*(arcsi n(c*x)^2-2+2*I*arcsin(c*x))*d/c^2/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*( I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*f*g*(arcsin(c*x)^2-2-2*I*arcsin(c*x))* d/c^2/(c^2*x^2-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*...
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm= "fricas")
Output:
integral(-(a^2*c^2*d*g^2*x^4 + 2*a^2*c^2*d*f*g*x^3 - 2*a^2*d*f*g*x - a^2*d *f^2 + (a^2*c^2*d*f^2 - a^2*d*g^2)*x^2 + (b^2*c^2*d*g^2*x^4 + 2*b^2*c^2*d* f*g*x^3 - 2*b^2*d*f*g*x - b^2*d*f^2 + (b^2*c^2*d*f^2 - b^2*d*g^2)*x^2)*arc sin(c*x)^2 + 2*(a*b*c^2*d*g^2*x^4 + 2*a*b*c^2*d*f*g*x^3 - 2*a*b*d*f*g*x - a*b*d*f^2 + (a*b*c^2*d*f^2 - a*b*d*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:
integrate((g*x+f)**2*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2,x)
Output:
Timed out
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm= "maxima")
Output:
1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a rcsin(c*x)/c)*a^2*f^2 + 1/48*a^2*g^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*( -c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/ 2)*arcsin(c*x)/c^3) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a^2*f*g/(c^2*d) + sqrt(d) *integrate(-((b^2*c^2*d*g^2*x^4 + 2*b^2*c^2*d*f*g*x^3 - 2*b^2*d*f*g*x - b^ 2*d*f^2 + (b^2*c^2*d*f^2 - b^2*d*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt (-c*x + 1))^2 + 2*(a*b*c^2*d*g^2*x^4 + 2*a*b*c^2*d*f*g*x^3 - 2*a*b*d*f*g*x - a*b*d*f^2 + (a*b*c^2*d*f^2 - a*b*d*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. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm= "giac")
Output:
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. \[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:
int((f + g*x)^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2),x)
Output:
int((f + g*x)^2*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(3/2), x)
\[ \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, d \left (90 \mathit {asin} \left (c x \right ) a^{2} c^{2} f^{2}+15 \mathit {asin} \left (c x \right ) a^{2} g^{2}-60 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} f^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} f g \,x^{4}-40 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} g^{2} x^{5}+150 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} f^{2} x +192 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} f g \,x^{2}+70 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} g^{2} x^{3}-96 \sqrt {-c^{2} x^{2}+1}\, a^{2} c f g -15 \sqrt {-c^{2} x^{2}+1}\, a^{2} c \,g^{2} x -480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) a b \,c^{5} g^{2}-960 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) a b \,c^{5} f g -480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{5} f^{2}+480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3} g^{2}+960 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{3} f g +480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) a b \,c^{3} f^{2}-240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5} g^{2}-480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{5} f g -240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{5} f^{2}+240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3} g^{2}+480 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{3} f g +240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c^{3} f^{2}+96 a^{2} c f g \right )}{240 c^{3}} \] Input:
int((g*x+f)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x))^2,x)
Output:
(sqrt(d)*d*(90*asin(c*x)*a**2*c**2*f**2 + 15*asin(c*x)*a**2*g**2 - 60*sqrt ( - c**2*x**2 + 1)*a**2*c**5*f**2*x**3 - 96*sqrt( - c**2*x**2 + 1)*a**2*c* *5*f*g*x**4 - 40*sqrt( - c**2*x**2 + 1)*a**2*c**5*g**2*x**5 + 150*sqrt( - c**2*x**2 + 1)*a**2*c**3*f**2*x + 192*sqrt( - c**2*x**2 + 1)*a**2*c**3*f*g *x**2 + 70*sqrt( - c**2*x**2 + 1)*a**2*c**3*g**2*x**3 - 96*sqrt( - c**2*x* *2 + 1)*a**2*c*f*g - 15*sqrt( - c**2*x**2 + 1)*a**2*c*g**2*x - 480*int(sqr t( - c**2*x**2 + 1)*asin(c*x)*x**4,x)*a*b*c**5*g**2 - 960*int(sqrt( - c**2 *x**2 + 1)*asin(c*x)*x**3,x)*a*b*c**5*f*g - 480*int(sqrt( - c**2*x**2 + 1) *asin(c*x)*x**2,x)*a*b*c**5*f**2 + 480*int(sqrt( - c**2*x**2 + 1)*asin(c*x )*x**2,x)*a*b*c**3*g**2 + 960*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x,x)*a* b*c**3*f*g + 480*int(sqrt( - c**2*x**2 + 1)*asin(c*x),x)*a*b*c**3*f**2 - 2 40*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**4,x)*b**2*c**5*g**2 - 480*in t(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**3,x)*b**2*c**5*f*g - 240*int(sqrt ( - c**2*x**2 + 1)*asin(c*x)**2*x**2,x)*b**2*c**5*f**2 + 240*int(sqrt( - c **2*x**2 + 1)*asin(c*x)**2*x**2,x)*b**2*c**3*g**2 + 480*int(sqrt( - c**2*x **2 + 1)*asin(c*x)**2*x,x)*b**2*c**3*f*g + 240*int(sqrt( - c**2*x**2 + 1)* asin(c*x)**2,x)*b**2*c**3*f**2 + 96*a**2*c*f*g))/(240*c**3)