Integrand size = 31, antiderivative size = 422 \[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {3 b f^2 g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {2 b g^3 x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}-\frac {3 f^2 g \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2 d}-\frac {2 g^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^4 d}-\frac {3 f g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {g^3 x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 c^2 d}-\frac {f^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \] Output:
-3*b*f^2*g*x*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-2/3*b*g^3*x*(-c^2*x ^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)-3/4*b*f*g^2*x^2*(-c^2*x^2+1)^(1/2)/c/ (-c^2*d*x^2+d)^(1/2)-1/9*b*g^3*x^3*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/ 2)-3*f^2*g*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2/d-2/3*g^3*(-c^2*d*x^ 2+d)^(1/2)*(a+b*arccos(c*x))/c^4/d-3/2*f*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*a rccos(c*x))/c^2/d-1/3*g^3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2/d -1/2*f^3*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/b/c/(-c^2*d*x^2+d)^(1/2)-3 /4*f*g^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/b/c^3/(-c^2*d*x^2+d)^(1/2)
Time = 0.96 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.81 \[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {18 b c \sqrt {d} f \left (2 c^2 f^2+3 g^2\right ) \left (-1+c^2 x^2\right ) \arccos (c x)^2-36 a c f \left (2 c^2 f^2+3 g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} g \left (-1+c^2 x^2\right ) \left (8 b c x \left (6 g^2+c^2 \left (27 f^2+g^2 x^2\right )\right )+12 a \sqrt {1-c^2 x^2} \left (4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )+27 b c f g \cos (2 \arccos (c x))\right )+6 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arccos (c x) \left (4 \sqrt {1-c^2 x^2} \left (2 g^2+c^2 \left (9 f^2+g^2 x^2\right )\right )+9 c f g \sin (2 \arccos (c x))\right )}{72 c^4 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \] Input:
Integrate[((f + g*x)^3*(a + b*ArcCos[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
(18*b*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*(-1 + c^2*x^2)*ArcCos[c*x]^2 - 36*a* c*f*(2*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x* Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*g*(-1 + c^2*x^2)* (8*b*c*x*(6*g^2 + c^2*(27*f^2 + g^2*x^2)) + 12*a*Sqrt[1 - c^2*x^2]*(4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2)) + 27*b*c*f*g*Cos[2*ArcCos[c*x]]) + 6 *b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcCos[c*x]*(4*Sqrt[1 - c^2*x^2]*(2*g^2 + c^2* (9*f^2 + g^2*x^2)) + 9*c*f*g*Sin[2*ArcCos[c*x]]))/(72*c^4*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])
Time = 0.88 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.62, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5263 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \left (\frac {(a+b \arccos (c x)) f^3}{\sqrt {1-c^2 x^2}}+\frac {3 g x (a+b \arccos (c x)) f^2}{\sqrt {1-c^2 x^2}}+\frac {3 g^2 x^2 (a+b \arccos (c x)) f}{\sqrt {1-c^2 x^2}}+\frac {g^3 x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right )dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {3 f g^2 (a+b \arccos (c x))^2}{4 b c^3}-\frac {3 f^2 g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {3 f g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {g^3 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {2 g^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^4}-\frac {f^3 (a+b \arccos (c x))^2}{2 b c}-\frac {2 b g^3 x}{3 c^3}-\frac {3 b f^2 g x}{c}-\frac {3 b f g^2 x^2}{4 c}-\frac {b g^3 x^3}{9 c}\right )}{\sqrt {d-c^2 d x^2}}\) |
Input:
Int[((f + g*x)^3*(a + b*ArcCos[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[1 - c^2*x^2]*((-3*b*f^2*g*x)/c - (2*b*g^3*x)/(3*c^3) - (3*b*f*g^2*x^ 2)/(4*c) - (b*g^3*x^3)/(9*c) - (3*f^2*g*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c* x]))/c^2 - (2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3*c^4) - (3*f*g^ 2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c^2) - (g^3*x^2*Sqrt[1 - c^2 *x^2]*(a + b*ArcCos[c*x]))/(3*c^2) - (f^3*(a + b*ArcCos[c*x])^2)/(2*b*c) - (3*f*g^2*(a + b*ArcCos[c*x])^2)/(4*b*c^3)))/Sqrt[d - c^2*d*x^2]
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 = 0.72 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.03
| method | result | size |
| default | \(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f \left (2 c^{2} f^{2}+3 g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right ) g^{3} \left (i+3 \arccos \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (4 i c^{2} f^{2}+4 \arccos \left (c x \right ) c^{2} f^{2}+i g^{2}+\arccos \left (c x \right ) g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (-4 i c^{2} f^{2}+4 \arccos \left (c x \right ) c^{2} f^{2}-i g^{2}+\arccos \left (c x \right ) g^{2}\right )}{8 c^{4} d \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 c +2 c^{2} x^{2}-1\right ) g^{3} \left (-i+3 \arccos \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \arccos \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \,g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{3} \cos \left (4 \arccos \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sin \left (4 \arccos \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\) | \(856\) |
| parts | \(a \left (\frac {f^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{3} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {3 f^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f \left (2 c^{2} f^{2}+3 g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right ) g^{3} \left (i+3 \arccos \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (4 i c^{2} f^{2}+4 \arccos \left (c x \right ) c^{2} f^{2}+i g^{2}+\arccos \left (c x \right ) g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (-4 i c^{2} f^{2}+4 \arccos \left (c x \right ) c^{2} f^{2}-i g^{2}+\arccos \left (c x \right ) g^{2}\right )}{8 c^{4} d \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 c +2 c^{2} x^{2}-1\right ) g^{3} \left (-i+3 \arccos \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \arccos \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \,g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{3} \cos \left (4 \arccos \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{3} \sin \left (4 \arccos \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,g^{2} \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\) | \(856\) |
Input:
int((g*x+f)^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBO SE)
Output:
a*(f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+g^3*(-1/ 3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+3*f*g^2*( -1/2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/ 2)*x/(-c^2*d*x^2+d)^(1/2)))-3*f^2*g/c^2/d*(-c^2*d*x^2+d)^(1/2))+b*(1/4*(-d *(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arccos(c*x)^2*f*( 2*c^2*f^2+3*g^2)+1/144*(-d*(c^2*x^2-1))^(1/2)*(2*c^2*x^2-1+2*I*(-c^2*x^2+1 )^(1/2)*c*x)*g^3*(I+3*arccos(c*x))/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^ (1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(4*I*c^2*f^2+4*arccos(c*x)*c^ 2*f^2+I*g^2+arccos(c*x)*g^2)/c^4/d/(c^2*x^2-1)-3/8*(-d*(c^2*x^2-1))^(1/2)* (-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*g*(-4*I*c^2*f^2+4*arccos(c*x)*c^2*f^ 2-I*g^2+arccos(c*x)*g^2)/c^4/d/(c^2*x^2-1)+1/144*(-d*(c^2*x^2-1))^(1/2)*(- 2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-1)*g^3*(-I+3*arccos(c*x))/c^4/d/(c^2* x^2-1)+3/8*(-d*(c^2*x^2-1))^(1/2)/c^2/d/(c^2*x^2-1)*f*g^2*arccos(c*x)*x-3/ 16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2-1/24* (-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*arccos(c*x)*g^3*cos(4*arccos(c*x) )+1/72*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*g^3*sin(4*arccos(c*x))-3/8 *(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*arccos(c*x)*cos(3*arccos(c *x))+3/16*(-d*(c^2*x^2-1))^(1/2)/c^3/d/(c^2*x^2-1)*f*g^2*sin(3*arccos(c*x) ))
\[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="f ricas")
Output:
integral(-(a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arccos(c*x))*sqrt(-c^2*d*x^2 + d)/(c^ 2*d*x^2 - d), x)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)**3*(a+b*acos(c*x))/(-c**2*d*x**2+d)**(1/2),x)
Output:
Exception raised: TypeError >> Invalid comparison of non-real zoo
\[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((g*x+f)^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="m axima")
Output:
-1/3*a*g^3*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4 *d)) - 3/2*a*f*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt (d))) + b*f^3*arccos(c*x)*arcsin(c*x)/(c*sqrt(d)) + 1/2*b*f^3*arcsin(c*x)^ 2/(c*sqrt(d)) - 3*b*f^2*g*x/(c*sqrt(d)) + a*f^3*arcsin(c*x)/(c*sqrt(d)) - 3*sqrt(-c^2*d*x^2 + d)*b*f^2*g*arccos(c*x)/(c^2*d) - 3*sqrt(-c^2*d*x^2 + d )*a*f^2*g/(c^2*d) - sqrt(d)*integrate((b*g^3*x^3 + 3*b*f*g^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^2*d*x^2 - d), x)
Exception generated. \[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((g*x+f)^3*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="g iac")
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. \[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int(((f + g*x)^3*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(1/2),x)
Output:
int(((f + g*x)^3*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {(f+g x)^3 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-3 \mathit {acos} \left (c x \right )^{2} b \,c^{3} f^{3}-18 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b \,c^{2} f^{2} g +6 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+9 \mathit {asin} \left (c x \right ) a c f \,g^{2}-18 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -9 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+6 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} g^{3}+18 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} f \,g^{2}-18 b \,c^{3} f^{2} g x}{6 \sqrt {d}\, c^{4}} \] Input:
int((g*x+f)^3*(a+b*acos(c*x))/(-c^2*d*x^2+d)^(1/2),x)
Output:
( - 3*acos(c*x)**2*b*c**3*f**3 - 18*sqrt( - c**2*x**2 + 1)*acos(c*x)*b*c** 2*f**2*g + 6*asin(c*x)*a*c**3*f**3 + 9*asin(c*x)*a*c*f*g**2 - 18*sqrt( - c **2*x**2 + 1)*a*c**2*f**2*g - 9*sqrt( - c**2*x**2 + 1)*a*c**2*f*g**2*x - 2 *sqrt( - c**2*x**2 + 1)*a*c**2*g**3*x**2 - 4*sqrt( - c**2*x**2 + 1)*a*g**3 + 6*int((acos(c*x)*x**3)/sqrt( - c**2*x**2 + 1),x)*b*c**4*g**3 + 18*int(( acos(c*x)*x**2)/sqrt( - c**2*x**2 + 1),x)*b*c**4*f*g**2 - 18*b*c**3*f**2*g *x)/(6*sqrt(d)*c**4)