\(\int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx\) [2]

Optimal result

Integrand size = 31, antiderivative size = 443 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=-\frac {2 b f g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c f^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {1-c^2 x^2}}+\frac {2 b c f g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {b c g^2 x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {2 f g \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2 d}-\frac {f^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {g^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{16 b c^3 \sqrt {1-c^2 x^2}} \] Output:

-2/3*b*f*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+1/4*b*c*f^2*x^2*(-c 
^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/16*b*g^2*x^2*(-c^2*d*x^2+d)^(1/2)/c 
/(-c^2*x^2+1)^(1/2)+2/9*b*c*f*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2 
)+1/16*b*c*g^2*x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/2*f^2*x*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccos(c*x))-1/8*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
os(c*x))/c^2+1/4*g^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))-2/3*f*g*(- 
c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/c^2/d-1/4*f^2*(-c^2*d*x^2+d)^(1/2)*(a 
+b*arccos(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)-1/16*g^2*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arccos(c*x))^2/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.72 \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {48 a c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (-1+c^2 x^2\right )+3 g^2 x \left (-1+2 c^2 x^2\right )\right )-144 a \sqrt {d} \left (4 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-64 b c f g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)-\cos (3 \arccos (c x))\right )+144 b c^2 f^2 \sqrt {d-c^2 d x^2} (\cos (2 \arccos (c x))+2 \arccos (c x) (-\arccos (c x)+\sin (2 \arccos (c x))))+9 b g^2 \sqrt {d-c^2 d x^2} \left (-8 \arccos (c x)^2+\cos (4 \arccos (c x))+4 \arccos (c x) \sin (4 \arccos (c x))\right )}{1152 c^3 \sqrt {1-c^2 x^2}} \] Input:

Integrate[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]
 

Output:

(48*a*c*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(12*c^2*f^2*x + 16*f*g*(-1 + 
 c^2*x^2) + 3*g^2*x*(-1 + 2*c^2*x^2)) - 144*a*Sqrt[d]*(4*c^2*f^2 + g^2)*Sq 
rt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] 
 - 64*b*c*f*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*(1 - c^2*x^2)^(3/2)*ArcCos[c 
*x] - Cos[3*ArcCos[c*x]]) + 144*b*c^2*f^2*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCo 
s[c*x]] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])) + 9*b*g^2*Sqr 
t[d - c^2*d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*ArcCos[c*x]] + 4*ArcCos[c*x]*Si 
n[4*ArcCos[c*x]]))/(1152*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.59, 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.

\(\displaystyle \int \sqrt {d-c^2 d x^2} (f+g x)^2 (a+b \arccos (c x)) \, dx\)

\(\Big \downarrow \) 5277

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int (f+g x)^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5263

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \left (\sqrt {1-c^2 x^2} (a+b \arccos (c x)) f^2+2 g x \sqrt {1-c^2 x^2} (a+b \arccos (c x)) f+g^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {g^2 (a+b \arccos (c x))^2}{16 b c^3}+\frac {1}{2} f^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {2 f g \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2}-\frac {g^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {f^2 (a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c f^2 x^2+\frac {2}{9} b c f g x^3-\frac {2 b f g x}{3 c}+\frac {1}{16} b c g^2 x^4-\frac {b g^2 x^2}{16 c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]
 

Output:

(Sqrt[d - c^2*d*x^2]*((-2*b*f*g*x)/(3*c) + (b*c*f^2*x^2)/4 - (b*g^2*x^2)/( 
16*c) + (2*b*c*f*g*x^3)/9 + (b*c*g^2*x^4)/16 + (f^2*x*Sqrt[1 - c^2*x^2]*(a 
 + b*ArcCos[c*x]))/2 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(8*c^ 
2) + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/4 - (2*f*g*(1 - c^2*x 
^2)^(3/2)*(a + b*ArcCos[c*x]))/(3*c^2) - (f^2*(a + b*ArcCos[c*x])^2)/(4*b* 
c) - (g^2*(a + b*ArcCos[c*x])^2)/(16*b*c^3)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5263

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))
 

rule 5277

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.20

method	result	size
default	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 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} \left (4 c^{2} f^{2}+g^{2}\right )}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) g^{2} \left (i+4 \arccos \left (c x \right )\right )}{256 c^{3} \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 \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) f g \left (i+3 \arccos \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) f^{2} \left (i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\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 ) f g \left (\arccos \left (c x \right )+i\right )}{4 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\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 ) f g \left (\arccos \left (c x \right )-i\right )}{4 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^{2} \left (-i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) f g \left (-i+3 \arccos \left (c x \right )\right )}{36 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 ) g^{2} \left (-i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(973\)
parts	\(a \left (f^{2} \left (\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (-\frac {x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )-\frac {2 f g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 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} \left (4 c^{2} f^{2}+g^{2}\right )}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) g^{2} \left (i+4 \arccos \left (c x \right )\right )}{256 c^{3} \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 \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) f g \left (i+3 \arccos \left (c x \right )\right )}{36 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) f^{2} \left (i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\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 ) f g \left (\arccos \left (c x \right )+i\right )}{4 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\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 ) f g \left (\arccos \left (c x \right )-i\right )}{4 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^{2} \left (-i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) f g \left (-i+3 \arccos \left (c x \right )\right )}{36 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 ) g^{2} \left (-i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(973\)

Input:

int((g*x+f)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBO 
SE)
 

Output:

a*(f^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)))+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))))-2/3*f*g/c^2/d*(-c^2*d*x^2+d)^(3/2))+b*(1/16*(-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*(4*c^2*f^2+ 
g^2)+1/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^( 
1/2)*x^4*c^4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*g^ 
2*(I+4*arccos(c*x))/c^3/(c^2*x^2-1)+1/36*(-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)*c*x+1)*f* 
g*(I+3*arccos(c*x))/c^2/(c^2*x^2-1)+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3 
-2*c*x+2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*f^2*(I+2*arcco 
s(c*x))/c/(c^2*x^2-1)-1/4*(-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*(arccos(c*x)+I)/c^2/(c^2*x^2-1)-1/4*(-d*(c^2*x^2-1))^(1/2) 
*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*f*g*(arccos(c*x)-I)/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^2*(-I+2*arccos(c*x))/c/(c^2*x^2-1)+1/36*(-d*(c 
^2*x^2-1))^(1/2)*(-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4+3*I*(-c^2*x^2+ 
1)^(1/2)*c*x-5*c^2*x^2+1)*f*g*(-I+3*arccos(c*x))/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)*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)*g^2*(-I+4*ar...
 

Fricas [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x, algorithm="f 
ricas")
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 
+ 2*b*f*g*x + b*f^2)*arccos(c*x)), x)
 

Sympy [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \] Input:

integrate((g*x+f)**2*(-c**2*d*x**2+d)**(1/2)*(a+b*acos(c*x)),x)
 

Output:

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))*(f + g*x)**2, x)
 

Maxima [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),x, algorithm="m 
axima")
 

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f^2 + 1/8*a*g^2*(sq 
rt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + sqrt(d)*ar 
csin(c*x)/c^3) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*f*g/(c^2*d) + sqrt(d)*integr 
ate((b*g^2*x^2 + 2*b*f*g*x + b*f^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(s 
qrt(c*x + 1)*sqrt(-c*x + 1), c*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((g*x+f)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x)),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
 

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \] Input:

int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(1/2),x)
 

Output:

int((f + g*x)^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int (f+g x)^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, \left (12 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+3 \mathit {asin} \left (c x \right ) a \,g^{2}+12 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f^{2} x +16 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} f g \,x^{2}+6 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} g^{2} x^{3}-16 \sqrt {-c^{2} x^{2}+1}\, a c f g -3 \sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x +24 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3} g^{2}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x d x \right ) b \,c^{3} f g +24 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) b \,c^{3} f^{2}+16 a c f g \right )}{24 c^{3}} \] Input:

int((g*x+f)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*acos(c*x)),x)
 

Output:

(sqrt(d)*(12*asin(c*x)*a*c**2*f**2 + 3*asin(c*x)*a*g**2 + 12*sqrt( - c**2* 
x**2 + 1)*a*c**3*f**2*x + 16*sqrt( - c**2*x**2 + 1)*a*c**3*f*g*x**2 + 6*sq 
rt( - c**2*x**2 + 1)*a*c**3*g**2*x**3 - 16*sqrt( - c**2*x**2 + 1)*a*c*f*g 
- 3*sqrt( - c**2*x**2 + 1)*a*c*g**2*x + 24*int(sqrt( - c**2*x**2 + 1)*acos 
(c*x)*x**2,x)*b*c**3*g**2 + 48*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x,x)*b 
*c**3*f*g + 24*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x)*b*c**3*f**2 + 16*a* 
c*f*g))/(24*c**3)