\(\int \frac {(f+g x)^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx\) [15]

Optimal result

Integrand size = 31, antiderivative size = 256 \[ \int \frac {(f+g x)^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^2 d}-\frac {g^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 c^2 d}-\frac {f^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}} \] Output:

-2*b*f*g*x*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-1/4*b*g^2*x^2*(-c^2*x 
^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-2*f*g*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos( 
c*x))/c^2/d-1/2*g^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2/d-1/2*f^2 
*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/b/c/(-c^2*d*x^2+d)^(1/2)-1/4*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)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.04 \[ \int \frac {(f+g x)^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2 b \sqrt {d} \left (2 c^2 f^2+g^2\right ) \left (-1+c^2 x^2\right ) \arccos (c x)^2-4 a \left (2 c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} g \left (-1+c^2 x^2\right ) \left (4 c \left (4 b c f x+a (4 f+g x) \sqrt {1-c^2 x^2}\right )+b g \cos (2 \arccos (c x))\right )+2 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arccos (c x) \left (8 c f \sqrt {1-c^2 x^2}+g \sin (2 \arccos (c x))\right )}{8 c^3 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.63, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[1 - c^2*x^2]*((-2*b*f*g*x)/c - (b*g^2*x^2)/(4*c) - (2*f*g*Sqrt[1 - c 
^2*x^2]*(a + b*ArcCos[c*x]))/c^2 - (g^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[ 
c*x]))/(2*c^2) - (f^2*(a + b*ArcCos[c*x])^2)/(2*b*c) - (g^2*(a + b*ArcCos[ 
c*x])^2)/(4*b*c^3)))/Sqrt[d - c^2*d*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 = 0.58 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.97

method	result	size
default	\(a \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f g \left (\arccos \left (c x \right )+i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f g \left (\arccos \left (c x \right )-i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(505\)
parts	\(a \left (\frac {f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+g^{2} \left (-\frac {x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {\arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )-\frac {2 f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} \left (2 c^{2} f^{2}+g^{2}\right )}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f g \left (\arccos \left (c x \right )+i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) f g \left (\arccos \left (c x \right )-i\right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) g^{2} \cos \left (3 \arccos \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \sin \left (3 \arccos \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}\right )\)	\(505\)

Input:

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

Output:

a*(f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+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)))-2*f*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*(2*c^2*f 
^2+g^2)-(-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*(a 
rccos(c*x)+I)/c^2/d/(c^2*x^2-1)-(-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/d/(c^2*x^2-1)+1/8*(-d*(c^2*x^2- 
1))^(1/2)/c^2/d/(c^2*x^2-1)*arccos(c*x)*g^2*x-1/16*(-d*(c^2*x^2-1))^(1/2)* 
(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*g^2-1/8*(-d*(c^2*x^2-1))^(1/2)/c^3/d/ 
(c^2*x^2-1)*arccos(c*x)*g^2*cos(3*arccos(c*x))+1/16*(-d*(c^2*x^2-1))^(1/2) 
/c^3/d/(c^2*x^2-1)*g^2*sin(3*arccos(c*x)))
 

Fricas [F]

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

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

Sympy [F(-2)]

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

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

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

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

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

Output:

-1/2*a*g^2*(sqrt(-c^2*d*x^2 + d)*x/(c^2*d) - arcsin(c*x)/(c^3*sqrt(d))) + 
b*f^2*arccos(c*x)*arcsin(c*x)/(c*sqrt(d)) + 1/2*b*f^2*arcsin(c*x)^2/(c*sqr 
t(d)) + b*g^2*integrate(x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(sq 
rt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d) - 2*b*f*g*x/(c*sqrt(d)) + a*f^2*ar 
csin(c*x)/(c*sqrt(d)) - 2*sqrt(-c^2*d*x^2 + d)*b*f*g*arccos(c*x)/(c^2*d) - 
 2*sqrt(-c^2*d*x^2 + d)*a*f*g/(c^2*d)
 

Giac [F(-2)]

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

integrate((g*x+f)^2*(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
 

Mupad [F(-1)]

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

\[ \int \frac {(f+g x)^2 (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\mathit {acos} \left (c x \right )^{2} b \,c^{2} f^{2}-4 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b c f g +2 \mathit {asin} \left (c x \right ) a \,c^{2} f^{2}+\mathit {asin} \left (c x \right ) a \,g^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a c f g -\sqrt {-c^{2} x^{2}+1}\, a c \,g^{2} x +2 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{3} g^{2}+4 a c f g -4 b \,c^{2} f g x}{2 \sqrt {d}\, c^{3}} \] Input:

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

Output:

( - acos(c*x)**2*b*c**2*f**2 - 4*sqrt( - c**2*x**2 + 1)*acos(c*x)*b*c*f*g 
+ 2*asin(c*x)*a*c**2*f**2 + asin(c*x)*a*g**2 - 4*sqrt( - c**2*x**2 + 1)*a* 
c*f*g - sqrt( - c**2*x**2 + 1)*a*c*g**2*x + 2*int((acos(c*x)*x**2)/sqrt( - 
 c**2*x**2 + 1),x)*b*c**3*g**2 + 4*a*c*f*g - 4*b*c**2*f*g*x)/(2*sqrt(d)*c* 
*3)