Integrand size = 33, antiderivative size = 1084 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 f^2 g \sqrt {d-c^2 d x^2}}{3 c^2}+\frac {52 b^2 g^3 \sqrt {d-c^2 d x^2}}{225 c^4}-\frac {1}{4} b^2 f^3 x \sqrt {d-c^2 d x^2}+\frac {3 b^2 f g^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {2 b^2 f^2 g \left (d-c^2 d x^2\right )^{3/2}}{9 c^2 d}+\frac {26 b^2 g^3 \left (d-c^2 d x^2\right )^{3/2}}{675 c^4 d}-\frac {2 b^2 g^3 \left (d-c^2 d x^2\right )^{5/2}}{125 c^4 d^2}+\frac {b^2 f^3 \sqrt {d-c^2 d x^2} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {3 b^2 f g^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b f^2 g x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c \sqrt {1-c^2 x^2}}+\frac {4 b g^3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{15 c^3 \sqrt {1-c^2 x^2}}-\frac {b c f^3 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {3 b f g^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {2 b c f^2 g x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b g^3 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{45 c \sqrt {1-c^2 x^2}}-\frac {3 b c f g^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {2 b c g^3 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{25 \sqrt {1-c^2 x^2}}-\frac {2 g^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^4}+\frac {1}{2} f^3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^2}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{5} g^3 x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {f^2 g \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c^2 d}+\frac {f^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {f g^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{8 b c^3 \sqrt {1-c^2 x^2}} \] Output:
3/64*b^2*f*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2-3/8*f*g^2*x*(-c^2*d*x^2+d)^(1/2) *(a+b*arcsin(c*x))^2/c^2+1/4*b^2*f^3*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/c/(- c^2*x^2+1)^(1/2)+1/6*f^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(-c^ 2*x^2+1)^(1/2)+3/8*b*f*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)-2/3*b*c*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/ (-c^2*x^2+1)^(1/2)+2/9*b^2*f^2*g*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*f*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)-3/64*b^2*f*g ^2*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/c^3/(-c^2*x^2+1)^(1/2)+4/15*b*g^3*x*(- c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c^3/(-c^2*x^2+1)^(1/2)-1/2*b*c*f^3*x^ 2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+2/45*b*g^3*x^3 *(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)-2/25*b*c*g^3* x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+26/675*b^2*g ^3*(-c^2*d*x^2+d)^(3/2)/c^4/d-2/125*b^2*g^3*(-c^2*d*x^2+d)^(5/2)/c^4/d^2+5 2/225*b^2*g^3*(-c^2*d*x^2+d)^(1/2)/c^4-1/4*b^2*f^3*x*(-c^2*d*x^2+d)^(1/2)- 2/15*g^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^4+1/2*f^3*x*(-c^2*d*x^ 2+d)^(1/2)*(a+b*arcsin(c*x))^2+1/5*g^3*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsi n(c*x))^2-f^2*g*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/c^2/d-3/8*b*c*f*g ^2*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+2*b*f^2*g *x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)+4/3*b^2*f^2 *g*(-c^2*d*x^2+d)^(1/2)/c^2-3/32*b^2*f*g^2*x^3*(-c^2*d*x^2+d)^(1/2)-1/1...
Time = 0.66 (sec) , antiderivative size = 708, normalized size of antiderivative = 0.65 \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {1}{2} f^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {3}{4} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2+\frac {1}{5} g^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {f^2 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c^2}+\frac {f^3 (a+b \arcsin (c x))^3}{6 b c}-\frac {2 b g^3 \left (15 a c^5 x^5+b \sqrt {1-c^2 x^2} \left (8+4 c^2 x^2+3 c^4 x^4\right )+15 b c^5 x^5 \arcsin (c x)\right )}{375 c^4}-\frac {2 b f^2 g \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{9 c^2}-\frac {b f^3 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )}{4 c}-\frac {3 b f g^2 \left (8 a c^4 x^4+b c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )+b \left (-3+8 c^4 x^4\right ) \arcsin (c x)\right )}{64 c^3}+\frac {g^3 \left (-9 a^2 \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+6 a b c x \left (6+c^2 x^2\right )+2 b^2 \sqrt {1-c^2 x^2} \left (20+c^2 x^2\right )+6 b \left (-3 a \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+b c x \left (6+c^2 x^2\right )\right ) \arcsin (c x)-9 b^2 \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right ) \arcsin (c x)^2\right )}{135 c^4}-\frac {f g^2 \left (6 c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 (a+b \arcsin (c x))^3}{b}-3 b \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )\right )}{16 c^3}\right )}{\sqrt {1-c^2 x^2}} \] Input:
Integrate[(f + g*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]
Output:
(Sqrt[d - c^2*d*x^2]*((f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 + (g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)^(3/2)*(a + b*Ar cSin[c*x])^2)/c^2 + (f^3*(a + b*ArcSin[c*x])^3)/(6*b*c) - (2*b*g^3*(15*a*c ^5*x^5 + b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + 15*b*c^5*x^5*Ar cSin[c*x]))/(375*c^4) - (2*b*f^2*g*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3 *a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x^2)*ArcSin[c*x]))/(9*c^2) - (b* f^3*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]) )/(4*c) - (3*b*f*g^2*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 ) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*c^3) + (g^3*(-9*a^2*Sqrt[1 - c^2* x^2]*(2 + c^2*x^2) + 6*a*b*c*x*(6 + c^2*x^2) + 2*b^2*Sqrt[1 - c^2*x^2]*(20 + c^2*x^2) + 6*b*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x ^2))*ArcSin[c*x] - 9*b^2*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]^2))/( 135*c^4) - (f*g^2*(6*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (2*(a + b*ArcSin[c*x])^3)/b - 3*b*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(16*c^3)))/Sqrt[1 - c^2*x^2]
Time = 1.90 (sec) , antiderivative size = 750, normalized size of antiderivative = 0.69, 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.
| \(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x)^3 (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5276 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (f+g x)^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 f^3+3 g x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 f^2+3 g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 f+g^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {f g^2 (a+b \arcsin (c x))^3}{8 b c^3}+\frac {1}{2} f^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {f^2 g \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c^2}-\frac {3 f g^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{8 c^2}+\frac {3}{4} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {g^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{15 c^2}+\frac {1}{5} g^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {2 g^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{15 c^4}-\frac {1}{2} b c f^3 x^2 (a+b \arcsin (c x))+\frac {f^3 (a+b \arcsin (c x))^3}{6 b c}-\frac {2}{3} b c f^2 g x^3 (a+b \arcsin (c x))+\frac {2 b f^2 g x (a+b \arcsin (c x))}{c}-\frac {3}{8} b c f g^2 x^4 (a+b \arcsin (c x))+\frac {3 b f g^2 x^2 (a+b \arcsin (c x))}{8 c}-\frac {2}{25} b c g^3 x^5 (a+b \arcsin (c x))+\frac {2 b g^3 x^3 (a+b \arcsin (c x))}{45 c}+\frac {4 a b g^3 x}{15 c^3}-\frac {3 b^2 f g^2 \arcsin (c x)}{64 c^3}+\frac {4 b^2 g^3 x \arcsin (c x)}{15 c^3}+\frac {b^2 f^3 \arcsin (c x)}{4 c}-\frac {1}{4} b^2 f^3 x \sqrt {1-c^2 x^2}+\frac {2 b^2 f^2 g \left (1-c^2 x^2\right )^{3/2}}{9 c^2}+\frac {4 b^2 f^2 g \sqrt {1-c^2 x^2}}{3 c^2}+\frac {3 b^2 f g^2 x \sqrt {1-c^2 x^2}}{64 c^2}-\frac {3}{32} b^2 f g^2 x^3 \sqrt {1-c^2 x^2}-\frac {2 b^2 g^3 \left (1-c^2 x^2\right )^{5/2}}{125 c^4}+\frac {26 b^2 g^3 \left (1-c^2 x^2\right )^{3/2}}{675 c^4}+\frac {52 b^2 g^3 \sqrt {1-c^2 x^2}}{225 c^4}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[(f + g*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]
Output:
(Sqrt[d - c^2*d*x^2]*((4*a*b*g^3*x)/(15*c^3) + (4*b^2*f^2*g*Sqrt[1 - c^2*x ^2])/(3*c^2) + (52*b^2*g^3*Sqrt[1 - c^2*x^2])/(225*c^4) - (b^2*f^3*x*Sqrt[ 1 - c^2*x^2])/4 + (3*b^2*f*g^2*x*Sqrt[1 - c^2*x^2])/(64*c^2) - (3*b^2*f*g^ 2*x^3*Sqrt[1 - c^2*x^2])/32 + (2*b^2*f^2*g*(1 - c^2*x^2)^(3/2))/(9*c^2) + (26*b^2*g^3*(1 - c^2*x^2)^(3/2))/(675*c^4) - (2*b^2*g^3*(1 - c^2*x^2)^(5/2 ))/(125*c^4) + (b^2*f^3*ArcSin[c*x])/(4*c) - (3*b^2*f*g^2*ArcSin[c*x])/(64 *c^3) + (4*b^2*g^3*x*ArcSin[c*x])/(15*c^3) + (2*b*f^2*g*x*(a + b*ArcSin[c* x]))/c - (b*c*f^3*x^2*(a + b*ArcSin[c*x]))/2 + (3*b*f*g^2*x^2*(a + b*ArcSi n[c*x]))/(8*c) - (2*b*c*f^2*g*x^3*(a + b*ArcSin[c*x]))/3 + (2*b*g^3*x^3*(a + b*ArcSin[c*x]))/(45*c) - (3*b*c*f*g^2*x^4*(a + b*ArcSin[c*x]))/8 - (2*b *c*g^3*x^5*(a + b*ArcSin[c*x]))/25 - (2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSi n[c*x])^2)/(15*c^4) + (f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 - (3*f*g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) - (g^3*x^2*Sqr t[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^2) + (3*f*g^2*x^3*Sqrt[1 - c^2 *x^2]*(a + b*ArcSin[c*x])^2)/4 + (g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[ c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/c^2 + (f^3* (a + b*ArcSin[c*x])^3)/(6*b*c) + (f*g^2*(a + b*ArcSin[c*x])^3)/(8*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 = 1.08 (sec) , antiderivative size = 2728, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2728\) |
| parts | \(\text {Expression too large to display}\) | \(2728\) |
Input:
int((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x,method=_RETURNVER BOSE)
Output:
a^2*(f^3*(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*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15 /d/c^4*(-c^2*d*x^2+d)^(3/2))+3*f*g^2*(-1/4*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/ 4/c^2*(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))))-f^2*g/c^2/d*(-c^2*d*x^2+d)^(3/2))+b^2*(-1/24*(- 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*f*(4 *c^2*f^2+3*g^2)+1/4000*(-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)*g^3*(10*I*arcsin(c*x)+25*arcsin(c*x)^2-2)/c^4/(c ^2*x^2-1)+3/512*(-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)*f*g^2*(4*I*arcsin(c*x)+8*arcsin(c*x)^2-1)/c^3/(c^2*x^2-1)+1/864*(-d*(c ^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*( -c^2*x^2+1)^(1/2)*x*c+1)*g*(72*I*arcsin(c*x)*c^2*f^2+108*arcsin(c*x)^2*c^2 *f^2+6*I*arcsin(c*x)*g^2+9*arcsin(c*x)^2*g^2-24*c^2*f^2-2*g^2)/c^4/(c^2*x^ 2-1)+1/16*(-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)*f^3*(2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)/c/(c ^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1) *g*(12*I*arcsin(c*x)*c^2*f^2+6*arcsin(c*x)^2*c^2*f^2+2*I*arcsin(c*x)*g^2+a rcsin(c*x)^2*g^2-12*c^2*f^2-2*g^2)/c^4/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1)...
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm= "fricas")
Output:
integral((a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2*g ^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a*b *g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arcsin(c*x))*sqrt(-c ^2*d*x^2 + d), x)
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{3}\, dx \] Input:
integrate((g*x+f)**3*(-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))**2,x)
Output:
Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2*(f + g*x)**3, x )
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2,x, algorithm= "maxima")
Output:
1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a^2*f^3 - 1/15*a^2*g^ 3*(3*(-c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(3/2)/(c^4*d) ) + 3/8*a^2*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x /(c^2*d) + sqrt(d)*arcsin(c*x)/c^3) - (-c^2*d*x^2 + d)^(3/2)*a^2*f^2*g/(c^ 2*d) + sqrt(d)*integrate(((b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt (-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
Exception generated. \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(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. \[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int {\left (f+g\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \] Input:
int((f + g*x)^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2),x)
Output:
int((f + g*x)^3*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2), x)
\[ \int (f+g x)^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, \left (60 \mathit {asin} \left (c x \right ) a^{2} c^{3} f^{3}+45 \mathit {asin} \left (c x \right ) a^{2} c f \,g^{2}+60 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} f^{3} x +120 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} f^{2} g \,x^{2}+90 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} f \,g^{2} x^{3}+24 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} g^{3} x^{4}-120 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f^{2} g -45 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} f \,g^{2} x -8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} g^{3} x^{2}-16 \sqrt {-c^{2} x^{2}+1}\, a^{2} g^{3}+240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) a b \,c^{4} g^{3}+720 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{4} f \,g^{2}+720 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{4} f^{2} g +240 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) a b \,c^{4} f^{3}+120 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4} g^{3}+360 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{4} f \,g^{2}+360 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{4} f^{2} g +120 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c^{4} f^{3}+120 a^{2} c^{2} f^{2} g +16 a^{2} g^{3}\right )}{120 c^{4}} \] Input:
int((g*x+f)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*asin(c*x))^2,x)
Output:
(sqrt(d)*(60*asin(c*x)*a**2*c**3*f**3 + 45*asin(c*x)*a**2*c*f*g**2 + 60*sq rt( - c**2*x**2 + 1)*a**2*c**4*f**3*x + 120*sqrt( - c**2*x**2 + 1)*a**2*c* *4*f**2*g*x**2 + 90*sqrt( - c**2*x**2 + 1)*a**2*c**4*f*g**2*x**3 + 24*sqrt ( - c**2*x**2 + 1)*a**2*c**4*g**3*x**4 - 120*sqrt( - c**2*x**2 + 1)*a**2*c **2*f**2*g - 45*sqrt( - c**2*x**2 + 1)*a**2*c**2*f*g**2*x - 8*sqrt( - c**2 *x**2 + 1)*a**2*c**2*g**3*x**2 - 16*sqrt( - c**2*x**2 + 1)*a**2*g**3 + 240 *int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3,x)*a*b*c**4*g**3 + 720*int(sqrt ( - c**2*x**2 + 1)*asin(c*x)*x**2,x)*a*b*c**4*f*g**2 + 720*int(sqrt( - c** 2*x**2 + 1)*asin(c*x)*x,x)*a*b*c**4*f**2*g + 240*int(sqrt( - c**2*x**2 + 1 )*asin(c*x),x)*a*b*c**4*f**3 + 120*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2 *x**3,x)*b**2*c**4*g**3 + 360*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**2 ,x)*b**2*c**4*f*g**2 + 360*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x,x)*b* *2*c**4*f**2*g + 120*int(sqrt( - c**2*x**2 + 1)*asin(c*x)**2,x)*b**2*c**4* f**3 + 120*a**2*c**2*f**2*g + 16*a**2*g**3))/(120*c**4)