\(\int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx\) [3]

Optimal result

Integrand size = 29, antiderivative size = 231 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=-\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))-\frac {g \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2 d}-\frac {f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b c \sqrt {1-c^2 x^2}} \] Output:

-1/3*b*g*x*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+1/4*b*c*f*x^2*(-c^2*d 
*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/9*b*c*g*x^3*(-c^2*d*x^2+d)^(1/2)/(-c^2* 
x^2+1)^(1/2)+1/2*f*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))-1/3*g*(-c^2*d* 
x^2+d)^(3/2)*(a+b*arccos(c*x))/c^2/d-1/4*f*(-c^2*d*x^2+d)^(1/2)*(a+b*arcco 
s(c*x))^2/b/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.95 \[ \int (f+g x) \sqrt {d-c^2 d x^2} (a+b \arccos (c x)) \, dx=\frac {12 a \sqrt {d-c^2 d x^2} \left (3 c^2 f x+2 g \left (-1+c^2 x^2\right )\right )-36 a c \sqrt {d} f \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {2 b g \sqrt {d-c^2 d x^2} \left (-9 c x-12 \left (1-c^2 x^2\right )^{3/2} \arccos (c x)+\cos (3 \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}+\frac {9 b c f \sqrt {d-c^2 d x^2} \left (-2 \arccos (c x)^2+\cos (2 \arccos (c x))+2 \arccos (c x) \sin (2 \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}}{72 c^2} \] Input:

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

Output:

(12*a*Sqrt[d - c^2*d*x^2]*(3*c^2*f*x + 2*g*(-1 + c^2*x^2)) - 36*a*c*Sqrt[d 
]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (2*b*g*Sq 
rt[d - c^2*d*x^2]*(-9*c*x - 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] + Cos[3*Arc 
Cos[c*x]]))/Sqrt[1 - c^2*x^2] + (9*b*c*f*Sqrt[d - c^2*d*x^2]*(-2*ArcCos[c* 
x]^2 + Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*Sin[2*ArcCos[c*x]]))/Sqrt[1 - c^ 
2*x^2])/(72*c^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.61, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5263

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(-1/3*(b*g*x)/c + (b*c*f*x^2)/4 + (b*c*g*x^3)/9 + (f* 
x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (g*(1 - c^2*x^2)^(3/2)*(a + b 
*ArcCos[c*x]))/(3*c^2) - (f*(a + b*ArcCos[c*x])^2)/(4*b*c)))/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 = 0.76 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.72

method	result	size
default	\(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) g \left (i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) f \left (i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(628\)
parts	\(\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) g \left (i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) f \left (i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) g \left (-i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(628\)

Input:

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

Output:

1/2*a*f*x*(-c^2*d*x^2+d)^(1/2)+1/2*a*f*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2 
)*x/(-c^2*d*x^2+d)^(1/2))-1/3*a*g/c^2/d*(-c^2*d*x^2+d)^(3/2)+b*(1/4*(-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+1/72*(-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)*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*(I+2*arccos(c*x))/c/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2) 
*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*g*(arccos(c*x)+I)/c^2/(c^2*x^2-1)-1/ 
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*(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*(-I+2*arccos(c*x))/c/(c^ 
2*x^2-1)+1/72*(-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)*g*(-I+3*arccos(c*x))/c^2/(c^ 
2*x^2-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((g*x+f)*(-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), x)
 

Maxima [F]

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

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

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a*f + sqrt(d)*integra 
te((b*g*x + b*f)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(- 
c*x + 1), c*x), x) - 1/3*(-c^2*d*x^2 + d)^(3/2)*a*g/(c^2*d)
 

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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