Integrand size = 29, antiderivative size = 370 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=-\frac {b d g x \sqrt {d-c^2 d x^2}}{5 c \sqrt {1-c^2 x^2}}+\frac {5 b c d f x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c d g x^3 \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b c^3 d f x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}-\frac {b c^3 d g x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{5 c^2}-\frac {3 d f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c \sqrt {1-c^2 x^2}} \]
3/8*d*f*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)+1/4*d*f*x*(-c^2*x^2+1)*(a +b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)-1/5*d*g*(-c^2*x^2+1)^2*(a+b*arccos(c* x))*(-c^2*d*x^2+d)^(1/2)/c^2-1/5*b*d*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+ 1)^(1/2)+5/16*b*c*d*f*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+2/15*b*c *d*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f*x^4*(-c^2* d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/25*b*c^3*d*g*x^5*(-c^2*d*x^2+d)^(1/2)/ (-c^2*x^2+1)^(1/2)-3/16*d*f*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/( -c^2*x^2+1)^(1/2)
Time = 2.25 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.91 \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\frac {-1800 b c d f \sqrt {d-c^2 d x^2} \arccos (c x)^2-3600 a c d^{3/2} f \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-d \sqrt {d-c^2 d x^2} \left (-1200 b c f \cos (2 \arccos (c x))-200 b g \cos (3 \arccos (c x))+3 \left (400 b c g x+80 a \sqrt {1-c^2 x^2} \left (8 g \left (-1+c^2 x^2\right )^2+5 c^2 f x \left (-5+2 c^2 x^2\right )\right )+25 b c f \cos (4 \arccos (c x))+8 b g \cos (5 \arccos (c x))\right )\right )+20 b d \sqrt {d-c^2 d x^2} \arccos (c x) \left (-100 g \sqrt {1-c^2 x^2}+160 c^2 g x^2 \sqrt {1-c^2 x^2}+120 c f \sin (2 \arccos (c x))-10 g \sin (3 \arccos (c x))-15 c f \sin (4 \arccos (c x))-6 g \sin (5 \arccos (c x))\right )}{9600 c^2 \sqrt {1-c^2 x^2}} \]
(-1800*b*c*d*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*c*d^(3/2)*f*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]*(-1200*b*c*f*Cos[2*ArcCos[c*x]] - 200*b*g*Cos[3*Arc Cos[c*x]] + 3*(400*b*c*g*x + 80*a*Sqrt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c^2*x^2)) + 25*b*c*f*Cos[4*ArcCos[c*x]] + 8*b*g*Cos[5* ArcCos[c*x]])) + 20*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-100*g*Sqrt[1 - c ^2*x^2] + 160*c^2*g*x^2*Sqrt[1 - c^2*x^2] + 120*c*f*Sin[2*ArcCos[c*x]] - 1 0*g*Sin[3*ArcCos[c*x]] - 15*c*f*Sin[4*ArcCos[c*x]] - 6*g*Sin[5*ArcCos[c*x] ]))/(9600*c^2*Sqrt[1 - c^2*x^2])
Time = 0.57 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^{3/2} (f+g x) (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int (f+g x) \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (f (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{3/2}+g x (a+b \arccos (c x)) \left (1-c^2 x^2\right )^{3/2}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {1}{4} f x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} f x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {g \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}-\frac {3 f (a+b \arccos (c x))^2}{16 b c}-\frac {1}{16} b c^3 f x^4-\frac {1}{25} b c^3 g x^5+\frac {5}{16} b c f x^2+\frac {2}{15} b c g x^3-\frac {b g x}{5 c}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*(-1/5*(b*g*x)/c + (5*b*c*f*x^2)/16 + (2*b*c*g*x^3)/ 15 - (b*c^3*f*x^4)/16 - (b*c^3*g*x^5)/25 + (3*f*x*Sqrt[1 - c^2*x^2]*(a + b *ArcCos[c*x]))/8 + (f*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 - (g*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(5*c^2) - (3*f*(a + b*ArcCos[c*x])^ 2)/(16*b*c)))/Sqrt[1 - c^2*x^2]
3.1.8.3.1 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 = 1.45 (sec) , antiderivative size = 1012, normalized size of antiderivative = 2.74
| method | result | size |
| default | \(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 i c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+16 c^{6} x^{6}-20 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-28 c^{4} x^{4}+5 i c x \sqrt {-c^{2} x^{2}+1}+13 c^{2} x^{2}-1\right ) g \left (i+5 \arccos \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) f \left (4 \arccos \left (c x \right )+i\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arccos \left (c x \right )-i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (11 i+45 \arccos \left (c x \right )\right ) \cos \left (4 \arccos \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x \sqrt {-c^{2} x^{2}+1}+i c^{2} x^{2}-i\right ) g \left (7 i+15 \arccos \left (c x \right )\right ) \sin \left (4 \arccos \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x \sqrt {-c^{2} x^{2}+1}+i c^{2} x^{2}-i\right ) f \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(1012\) |
| parts | \(\frac {a f x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a f d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a f \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5 c^{2} d}+b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f d}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 i c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+16 c^{6} x^{6}-20 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-28 c^{4} x^{4}+5 i c x \sqrt {-c^{2} x^{2}+1}+13 c^{2} x^{2}-1\right ) g \left (i+5 \arccos \left (c x \right )\right ) d}{800 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}-8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) f \left (4 \arccos \left (c x \right )+i\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arccos \left (c x \right )-i\right ) d}{16 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right ) d}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right ) d}{96 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (11 i+45 \arccos \left (c x \right )\right ) \cos \left (4 \arccos \left (c x \right )\right ) d}{1200 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x \sqrt {-c^{2} x^{2}+1}+i c^{2} x^{2}-i\right ) g \left (7 i+15 \arccos \left (c x \right )\right ) \sin \left (4 \arccos \left (c x \right )\right ) d}{600 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) f \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c x \sqrt {-c^{2} x^{2}+1}+i c^{2} x^{2}-i\right ) f \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 c \left (c^{2} x^{2}-1\right )}\right )\) | \(1012\) |
1/4*a*f*x*(-c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a*f*d^ 2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/5*a*g/c^2/d *(-c^2*d*x^2+d)^(5/2)+b*(3/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/ (c^2*x^2-1)*arccos(c*x)^2*f*d-1/800*(-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*(I+5*arccos(c*x))*d/c^2/(c^2*x^2- 1)-1/256*(-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*(4 *arccos(c*x)+I)*d/c/(c^2*x^2-1)-1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1 )^(1/2)*x*c+c^2*x^2-1)*g*(arccos(c*x)+I)*d/c^2/(c^2*x^2-1)-1/16*(-d*(c^2*x ^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(arccos(c*x)-I)*d/c^2/ (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*(-I+2*arccos(c*x))*d/c/(c^2*x^2-1)+ 1/96*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^ (1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*g*(-I+3*arccos(c*x))*d/c^2/(c^2*x^2-1) -1/1200*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*g*(11* I+45*arccos(c*x))*cos(4*arccos(c*x))*d/c^2/(c^2*x^2-1)-1/600*(-d*(c^2*x^2- 1))^(1/2)*(c*x*(-c^2*x^2+1)^(1/2)+I*c^2*x^2-I)*g*(7*I+15*arccos(c*x))*sin( 4*arccos(c*x))*d/c^2/(c^2*x^2-1)-3/256*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*( -c^2*x^2+1)^(1/2)*x*c-1)*f*(5*I+12*arccos(c*x))*cos(3*arccos(c*x))*d/c/...
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
integral(-(a*c^2*d*g*x^3 + a*c^2*d*f*x^2 - a*d*g*x - a*d*f + (b*c^2*d*g*x^ 3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), x)
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
\[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \]
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 - 1/5*(-c^2*d*x^2 + d)^(5/2)*a*g/(c^2*d) + sqrt(d)*integ rate(-(b*c^2*d*g*x^3 + b*c^2*d*f*x^2 - b*d*g*x - b*d*f)*sqrt(c*x + 1)*sqrt (-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)
Exception generated. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]