Integrand size = 31, antiderivative size = 959 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=-\frac {3 b d f^2 g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}-\frac {2 b d g^3 x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b c d f^3 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {3 b d f g^2 x^2 \sqrt {d-c^2 d x^2}}{32 c \sqrt {1-c^2 x^2}}+\frac {2 b c d f^2 g x^3 \sqrt {d-c^2 d x^2}}{5 \sqrt {1-c^2 x^2}}-\frac {b d g^3 x^3 \sqrt {d-c^2 d x^2}}{105 c \sqrt {1-c^2 x^2}}-\frac {b c^3 d f^3 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {7 b c d f g^2 x^4 \sqrt {d-c^2 d x^2}}{32 \sqrt {1-c^2 x^2}}-\frac {3 b c^3 d f^2 g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {8 b c d g^3 x^5 \sqrt {d-c^2 d x^2}}{175 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f g^2 x^6 \sqrt {d-c^2 d x^2}}{12 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g^3 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^3 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{16 c^2}+\frac {3}{8} d f g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{4} d f^3 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{2} d f g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {3 d f^2 g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^2}-\frac {d g^3 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^4}+\frac {d g^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{7 c^4}-\frac {3 d f^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}}-\frac {3 d f g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{32 b c^3 \sqrt {1-c^2 x^2}} \]
3/8*d*f^3*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-3/16*d*f*g^2*x*(a+b*arc cos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+3/8*d*f*g^2*x^3*(a+b*arccos(c*x))*(-c^2 *d*x^2+d)^(1/2)+1/4*d*f^3*x*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^ (1/2)+1/2*d*f*g^2*x^3*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)- 3/5*d*f^2*g*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2-1/5* d*g^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4+1/7*d*g^3* (-c^2*x^2+1)^3*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-3/5*b*d*f^2*g*x* (-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/35*b*d*g^3*x*(-c^2*d*x^2+d)^(1 /2)/c^3/(-c^2*x^2+1)^(1/2)+5/16*b*c*d*f^3*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x ^2+1)^(1/2)-3/32*b*d*f*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+2 /5*b*c*d*f^2*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/105*b*d*g^3*x ^3*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^3*x^4*(-c^2*d* x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+7/32*b*c*d*f*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)-3/25*b*c^3*d*f^2*g*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1 )^(1/2)+8/175*b*c*d*g^3*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/12*b *c^3*d*f*g^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/49*b*c^3*d*g^3* x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-3/16*d*f^3*(a+b*arccos(c*x))^2 *(-c^2*d*x^2+d)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-3/32*d*f*g^2*(a+b*arccos(c*x) )^2*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)
Time = 5.61 (sec) , antiderivative size = 910, normalized size of antiderivative = 0.95 \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {-88200 b c d f \left (2 c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2} \arccos (c x)^2-176400 a c d^{3/2} f \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-d \sqrt {d-c^2 d x^2} \left (352800 b c^3 f^2 g x+44100 b c g^3 x+564480 a c^2 f^2 g \sqrt {1-c^2 x^2}+53760 a g^3 \sqrt {1-c^2 x^2}-588000 a c^4 f^3 x \sqrt {1-c^2 x^2}+176400 a c^2 f g^2 x \sqrt {1-c^2 x^2}-1128960 a c^4 f^2 g x^2 \sqrt {1-c^2 x^2}+26880 a c^2 g^3 x^2 \sqrt {1-c^2 x^2}+235200 a c^6 f^3 x^3 \sqrt {1-c^2 x^2}-823200 a c^4 f g^2 x^3 \sqrt {1-c^2 x^2}+564480 a c^6 f^2 g x^4 \sqrt {1-c^2 x^2}-215040 a c^4 g^3 x^4 \sqrt {1-c^2 x^2}+470400 a c^6 f g^2 x^5 \sqrt {1-c^2 x^2}+134400 a c^6 g^3 x^6 \sqrt {1-c^2 x^2}-7350 b c f \left (16 c^2 f^2+3 g^2\right ) \cos (2 \arccos (c x))-4900 b g \left (12 c^2 f^2+g^2\right ) \cos (3 \arccos (c x))+7350 b c^3 f^3 \cos (4 \arccos (c x))-11025 b c f g^2 \cos (4 \arccos (c x))+7056 b c^2 f^2 g \cos (5 \arccos (c x))-588 b g^3 \cos (5 \arccos (c x))+2450 b c f g^2 \cos (6 \arccos (c x))+300 b g^3 \cos (7 \arccos (c x))\right )+140 b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (-4200 c^2 f^2 g \sqrt {1-c^2 x^2}+416 g^3 \sqrt {1-c^2 x^2}+6720 c^4 f^2 g x^2 \sqrt {1-c^2 x^2}-1256 c^2 g^3 x^2 \sqrt {1-c^2 x^2}+864 g^3 \left (1-c^2 x^2\right )^{3/2} \cos (2 \arccos (c x))+120 g^3 \left (1-c^2 x^2\right )^{3/2} \cos (4 \arccos (c x))+1680 c^3 f^3 \sin (2 \arccos (c x))+315 c f g^2 \sin (2 \arccos (c x))-420 c^2 f^2 g \sin (3 \arccos (c x))+140 g^3 \sin (3 \arccos (c x))-210 c^3 f^3 \sin (4 \arccos (c x))+315 c f g^2 \sin (4 \arccos (c x))-252 c^2 f^2 g \sin (5 \arccos (c x))+84 g^3 \sin (5 \arccos (c x))-105 c f g^2 \sin (6 \arccos (c x))\right )}{940800 c^4 \sqrt {1-c^2 x^2}} \]
(-88200*b*c*d*f*(2*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 1764 00*a*c*d^(3/2)*f*(2*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(352800*b*c^ 3*f^2*g*x + 44100*b*c*g^3*x + 564480*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 53760 *a*g^3*Sqrt[1 - c^2*x^2] - 588000*a*c^4*f^3*x*Sqrt[1 - c^2*x^2] + 176400*a *c^2*f*g^2*x*Sqrt[1 - c^2*x^2] - 1128960*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] + 26880*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 235200*a*c^6*f^3*x^3*Sqrt[1 - c ^2*x^2] - 823200*a*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2] + 564480*a*c^6*f^2*g*x^ 4*Sqrt[1 - c^2*x^2] - 215040*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 470400*a*c^ 6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 134400*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] - 7 350*b*c*f*(16*c^2*f^2 + 3*g^2)*Cos[2*ArcCos[c*x]] - 4900*b*g*(12*c^2*f^2 + g^2)*Cos[3*ArcCos[c*x]] + 7350*b*c^3*f^3*Cos[4*ArcCos[c*x]] - 11025*b*c*f *g^2*Cos[4*ArcCos[c*x]] + 7056*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] - 588*b*g^3* Cos[5*ArcCos[c*x]] + 2450*b*c*f*g^2*Cos[6*ArcCos[c*x]] + 300*b*g^3*Cos[7*A rcCos[c*x]]) + 140*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-4200*c^2*f^2*g*Sq rt[1 - c^2*x^2] + 416*g^3*Sqrt[1 - c^2*x^2] + 6720*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 1256*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 864*g^3*(1 - c^2*x^2)^(3/2 )*Cos[2*ArcCos[c*x]] + 120*g^3*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 16 80*c^3*f^3*Sin[2*ArcCos[c*x]] + 315*c*f*g^2*Sin[2*ArcCos[c*x]] - 420*c^2*f ^2*g*Sin[3*ArcCos[c*x]] + 140*g^3*Sin[3*ArcCos[c*x]] - 210*c^3*f^3*Sin[...
Time = 1.16 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5277, 5263, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x)^3 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x)^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) f^3+3 g x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) f^2+3 g^2 x^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) f+g^3 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (-\frac {3 f g^2 (a+b \arccos (c x))^2}{32 b c^3}+\frac {1}{4} f^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f^3 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3 f^2 g \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}-\frac {3 f g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{16 c^2}+\frac {1}{2} f g^2 x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {g^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {g^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {3 f^3 (a+b \arccos (c x))^2}{16 b c}-\frac {1}{16} b c^3 f^3 x^4-\frac {3}{25} b c^3 f^2 g x^5-\frac {1}{12} b c^3 f g^2 x^6-\frac {1}{49} b c^3 g^3 x^7-\frac {2 b g^3 x}{35 c^3}+\frac {5}{16} b c f^3 x^2+\frac {2}{5} b c f^2 g x^3-\frac {3 b f^2 g x}{5 c}+\frac {7}{32} b c f g^2 x^4-\frac {3 b f g^2 x^2}{32 c}+\frac {8}{175} b c g^3 x^5-\frac {b g^3 x^3}{105 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*((-3*b*f^2*g*x)/(5*c) - (2*b*g^3*x)/(35*c^3) + (5*b *c*f^3*x^2)/16 - (3*b*f*g^2*x^2)/(32*c) + (2*b*c*f^2*g*x^3)/5 - (b*g^3*x^3 )/(105*c) - (b*c^3*f^3*x^4)/16 + (7*b*c*f*g^2*x^4)/32 - (3*b*c^3*f^2*g*x^5 )/25 + (8*b*c*g^3*x^5)/175 - (b*c^3*f*g^2*x^6)/12 - (b*c^3*g^3*x^7)/49 + ( 3*f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/8 - (3*f*g^2*x*Sqrt[1 - c^2 *x^2]*(a + b*ArcCos[c*x]))/(16*c^2) + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/8 + (f^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 + (f *g^2*x^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/2 - (3*f^2*g*(1 - c^2*x^ 2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^2) - (g^3*(1 - c^2*x^2)^(5/2)*(a + b*Ar cCos[c*x]))/(5*c^4) + (g^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/(7*c^4 ) - (3*f^3*(a + b*ArcCos[c*x])^2)/(16*b*c) - (3*f*g^2*(a + b*ArcCos[c*x])^ 2)/(32*b*c^3)))/Sqrt[1 - c^2*x^2]
3.1.6.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcCos[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_.) + ArcCos[(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*ArcCos[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 = 1.67 (sec) , antiderivative size = 2146, normalized size of antiderivative = 2.24
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2146\) |
| parts | \(\text {Expression too large to display}\) | \(2146\) |
a*(f^3*(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*(-1/7*x^ 2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+3*f*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)))))-3/5*f^2*g/c^2/d*(-c^2*d*x^2+d)^(5/2))+b*(3/32*(-d*(c^ 2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arccos(c*x)^2*f*(2*c^2* f^2+g^2)*d-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+ 64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^ (1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*g^3*(I+ 7*arccos(c*x))*d/c^4/(c^2*x^2-1)-1/768*(-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)*f* g^2*(I+6*arccos(c*x))*d/c^3/(c^2*x^2-1)-1/3200*(-d*(c^2*x^2-1))^(1/2)*(16* I*c^5*x^5*(-c^2*x^2+1)^(1/2)+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28 *c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*g*(60*arccos(c*x)*c^2*f^ 2+12*I*f^2*c^2-5*arccos(c*x)*g^2-I*g^2)*d/c^4/(c^2*x^2-1)-1/512*(-d*(c^2*x ^2-1))^(1/2)*(8*I*(-c^2*x^2+1)^(1/2)*c^4*x^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*(2*I*c^2*f^2+8*arccos (c*x)*c^2*f^2-3*I*g^2-12*arccos(c*x)*g^2)*d/c^3/(c^2*x^2-1)+1/384*(-d*(...
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
integral(-(a*c^2*d*g^3*x^5 + 3*a*c^2*d*f*g^2*x^4 - 3*a*d*f^2*g*x - a*d*f^3 + (3*a*c^2*d*f^2*g - a*d*g^3)*x^3 + (a*c^2*d*f^3 - 3*a*d*f*g^2)*x^2 + (b* c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2*g*x - b*d*f^3 + (3*b*c^2*d *f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g^2)*x^2)*arccos(c*x))*sqrt (-c^2*d*x^2 + d), x)
Timed out. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \]
\[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
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*f^3 - 1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2 *d*x^2 + d)^(5/2)/(c^4*d))*a*g^3 + 1/16*a*f*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) - 3/5*(-c^2*d*x^2 + d)^(5/2)*a*f^2*g/(c^2*d ) + sqrt(d)*integrate(-(b*c^2*d*g^3*x^5 + 3*b*c^2*d*f*g^2*x^4 - 3*b*d*f^2* g*x - b*d*f^3 + (3*b*c^2*d*f^2*g - b*d*g^3)*x^3 + (b*c^2*d*f^3 - 3*b*d*f*g ^2)*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1) , c*x), x)
Exception generated. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]