\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx\) [5]

Optimal result

Integrand size = 31, antiderivative size = 851 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \] Output:

-a*(-c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/g/(g* 
x+f)+1/2*b*c^3*f^2*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)^2/g^2/(c^2*f^2-g^2)/(- 
c^2*x^2+1)^(1/2)-1/2*(c^2*f*x+g)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^ 
2/b/c/(c^2*f^2-g^2)/(g*x+f)^2/(-c^2*x^2+1)^(1/2)-1/2*(-c^2*x^2+1)^(1/2)*(- 
c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/(g*x+f)^2-a*c^3*f^2*(-c^2*d*x^2 
+d)^(1/2)*arcsin(c*x)/g^2/(c^2*f^2-g^2)/(-c^2*x^2+1)^(1/2)+a*c^2*f*(-c^2*d 
*x^2+d)^(1/2)*arctan((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))/g 
^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)+I*b*c^2*f*(-c^2*d*x^2+d)^(1/2)*a 
rccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^ 
2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b*c^2*f*(-c^2*d*x^2+d)^(1/2)*ar 
ccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2 
/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)-b*c*(-c^2*d*x^2+d)^(1/2)*ln(g*x+f) 
/g^2/(-c^2*x^2+1)^(1/2)+b*c^2*f*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+I*(-c 
^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/(c^2*f^2-g^2)^(1/2)/(-c^ 
2*x^2+1)^(1/2)-b*c^2*f*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+I*(-c^2*x^2+1) 
^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^ 
(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 6.84 (sec) , antiderivative size = 1116, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*((2*a*g*Sqrt[d - c^2*d*x^2])/(f + g*x) - 2*a*c*Sqrt[d]*ArcTan[(c*x*Sq 
rt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (2*a*c^2*Sqrt[d]*f*Log[f + 
g*x])/Sqrt[-(c^2*f^2) + g^2] + (2*a*c^2*Sqrt[d]*f*Log[d*(g + c^2*f*x) + Sq 
rt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[-(c^2*f^2) + g^2] 
+ b*c*Sqrt[d - c^2*d*x^2]*((2*g*ArcCos[c*x])/(c*f + c*g*x) - ArcCos[c*x]^2 
/Sqrt[1 - c^2*x^2] + (2*Log[1 + (g*x)/f])/Sqrt[1 - c^2*x^2] + (2*c*f*(2*Ar 
cCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 
 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2 
*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[ 
c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[ 
c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^(( 
I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I 
)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTan 
h[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2 
)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)]) 
] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/ 
Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) 
+ g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Ta 
n[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Ta 
n[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g ...
 

Rubi [A] (verified)

Time = 2.82 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5277, 5265, 27, 5255, 27, 5299, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5265

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5255

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5299

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}-\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arccos (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}+\frac {b c \left (\frac {a c^3 f^2 \arcsin (c x)}{g^2}-\frac {a c^2 f \sqrt {c^2 f^2-g^2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g (f+g x)}-\frac {b c^3 f^2 \arccos (c x)^2}{2 g^2}-\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {i b c^2 f \arccos (c x) \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {i b c^2 f \arccos (c x) \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2}+\frac {b \sqrt {1-c^2 x^2} \arccos (c x) (c f-g) (c f+g)}{g (f+g x)}-b c \left (1-\frac {c^2 f^2}{g^2}\right ) \log (f+g x)\right )}{c^2 f^2-g^2}}{b c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/(b*c*(f + 
 g*x)^2) - (((g + c^2*f*x)^2*(a + b*ArcCos[c*x])^2)/(2*(c^2*f^2 - g^2)*(f 
+ g*x)^2) + (b*c*((a*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2])/(g*(f + g*x)) 
+ (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/(g*(f + g*x)) - (b 
*c^3*f^2*ArcCos[c*x]^2)/(2*g^2) + (a*c^3*f^2*ArcSin[c*x])/g^2 - (a*c^2*f*S 
qrt[c^2*f^2 - g^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2* 
x^2])])/g^2 - (I*b*c^2*f*Sqrt[c^2*f^2 - g^2]*ArcCos[c*x]*Log[1 + (E^(I*Arc 
Cos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 + (I*b*c^2*f*Sqrt[c^2*f^2 - 
 g^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2] 
)])/g^2 - b*c*(1 - (c^2*f^2)/g^2)*Log[f + g*x] - (b*c^2*f*Sqrt[c^2*f^2 - g 
^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g^2 
+ (b*c^2*f*Sqrt[c^2*f^2 - g^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + S 
qrt[c^2*f^2 - g^2]))])/g^2))/(c^2*f^2 - g^2))/(b*c)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 5255

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) 
 + (g_.)*(x_))^(p_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^p*(d + e*x)^ 
m, x]}, Simp[(a + b*ArcCos[c*x])^n   u, x] + Simp[b*c*n   Int[SimplifyInteg 
rand[u*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x], x]] /; Free 
Q[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && Lt 
Q[m + p + 1, 0]
 

rule 5265

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[ 
(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f + g*x)^m)*(d + e*x^2)*((a + b* 
ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] + Simp[1/(b*c*Sqrt[d]*(n + 
1))   Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcC 
os[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + 
e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
 

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]
 

rule 5299

Int[(ArcCos[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x]) 
^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt 
Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1938 vs. \(2 (813 ) = 1626\).

Time = 0.87 (sec) , antiderivative size = 1939, normalized size of antiderivative = 2.28

method	result	size
default	\(\text {Expression too large to display}\)	\(1939\)
parts	\(\text {Expression too large to display}\)	\(1939\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\frac {\sqrt {d}\, \left (-\mathit {asin} \left (c x \right ) a \,c^{3} f^{3}-\mathit {asin} \left (c x \right ) a \,c^{3} f^{2} g x +\mathit {asin} \left (c x \right ) a c f \,g^{2}+\mathit {asin} \left (c x \right ) a c \,g^{3} x +2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}+2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f g x -\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +\sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{3} g^{2}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{2} g^{3} x -\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b f \,g^{4}-\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,g^{5} x \right )}{g^{2} \left (c^{2} f^{2} g x +c^{2} f^{3}-g^{3} x -f \,g^{2}\right )} \] Input:

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

Output:

(sqrt(d)*( - asin(c*x)*a*c**3*f**3 - asin(c*x)*a*c**3*f**2*g*x + asin(c*x) 
*a*c*f*g**2 + asin(c*x)*a*c*g**3*x + 2*sqrt(c**2*f**2 - g**2)*atan((tan(as 
in(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f**2 + 2*sqrt(c**2*f**2 
 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f* 
g*x - sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g + sqrt( - c**2*x**2 + 1)*a*g**3 
 + int((sqrt( - c**2*x**2 + 1)*acos(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)* 
b*c**2*f**3*g**2 + int((sqrt( - c**2*x**2 + 1)*acos(c*x))/(f**2 + 2*f*g*x 
+ g**2*x**2),x)*b*c**2*f**2*g**3*x - int((sqrt( - c**2*x**2 + 1)*acos(c*x) 
)/(f**2 + 2*f*g*x + g**2*x**2),x)*b*f*g**4 - int((sqrt( - c**2*x**2 + 1)*a 
cos(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)*b*g**5*x))/(g**2*(c**2*f**3 + c* 
*2*f**2*g*x - f*g**2 - g**3*x))