\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arccos (c x))}{f+g x} \, dx\) [9]

Optimal result

Integrand size = 31, antiderivative size = 1053 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}+\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 g^2}+\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 g}-\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c g^4 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c g^2 (f+g x)}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}} \] Output:

-a*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/g^3+1/3*b*c*d*x*(-c^2*d*x^2+d)^( 
1/2)/g/(-c^2*x^2+1)^(1/2)-b*c*d*(c*f-g)*(c*f+g)*x*(-c^2*d*x^2+d)^(1/2)/g^3 
/(-c^2*x^2+1)^(1/2)+1/4*b*c^3*d*f*x^2*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1 
)^(1/2)-1/9*b*c^3*d*x^3*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)-b*d*(c*f 
-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/g^3+1/2*c^2*d*f*x*(-c^2*d*x^2 
+d)^(1/2)*(a+b*arccos(c*x))/g^2+1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x)) 
/g-1/4*c*d*f*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/g^2/(-c^2*x^2+1)^( 
1/2)+1/2*c*d*(c*f-g)*(c*f+g)*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/ 
g^3/(-c^2*x^2+1)^(1/2)+1/2*d*(c^2*f^2-g^2)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cos(c*x))^2/b/c/g^4/(g*x+f)/(-c^2*x^2+1)^(1/2)+1/2*d*(c*f-g)*(c*f+g)*(-c^2 
*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/g^2/(g*x+f)+a*d 
*(c^2*f^2-g^2)^(3/2)*(-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^4/(-c^2*x^2+1)^(1/2)-I*b*d*(c^2*f^2-g^2)^(3/2 
)*(-c^2*d*x^2+d)^(1/2)*arccos(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^4/(-c^2*x^2+1)^(1/2)+I*b*d*(c^2*f^2-g^2)^(3/2)*(-c 
^2*d*x^2+d)^(1/2)*arccos(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^4/(-c^2*x^2+1)^(1/2)+b*d*(c^2*f^2-g^2)^(3/2)*(-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^4/(-c^2*x^2+1)^(1/2)-b*d*(c^2*f^2-g^2)^(3/2)*(-c^2*d*x^2+d)^(1/2)*po 
lylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^4/(-...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3034\) vs. \(2(1053)=2106\).

Time = 11.21 (sec) , antiderivative size = 3034, normalized size of antiderivative = 2.88 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]
 

Output:

Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f 
*x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) + (a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*Ar 
cTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4) + 
(a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2* 
f^2) + g^2)^(3/2)*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqr 
t[-(d*(-1 + c^2*x^2))]])/g^4 - (b*d*Sqrt[d*(1 - c^2*x^2)]*((-2*c*g*x)/Sqrt 
[1 - c^2*x^2] - 2*g*ArcCos[c*x] + (c*f*ArcCos[c*x]^2)/Sqrt[1 - c^2*x^2] + 
(2*(-(c*f) + g)*(c*f + g)*(2*ArcCos[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)*Ar 
cTanh[((-(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 + c*g*x] 
)] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/S 
qrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(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])/(Sqr 
t[2]*Sqrt[g]*Sqrt[c*f + c*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]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)...
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 700, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5277, 5267, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5267

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) \left (c^2 f^2-g^2\right ) (a+b \arccos (c x))^2}{2 b c g^2 (f+g x)}+\frac {\left (c^2 f^2-g^2\right )^2 (a+b \arccos (c x))^2}{2 b c g^4 (f+g x)}+\frac {c x \left (c^2 f^2-g^2\right ) (a+b \arccos (c x))^2}{2 b g^3}+\frac {c^2 f x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 g^2}+\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 g}-\frac {c f (a+b \arccos (c x))^2}{4 b g^2}+\frac {a \left (c^2 f^2-g^2\right )^{3/2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g^3}+\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {i b \arccos (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {i b \arccos (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^4}-\frac {b \sqrt {1-c^2 x^2} \arccos (c x) (c f-g) (c f+g)}{g^3}+\frac {b c^3 f x^2}{4 g^2}-\frac {b c^3 x^3}{9 g}-\frac {b c x \left (c^2 f^2-g^2\right )}{g^3}+\frac {b c x}{3 g}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*((b*c*x)/(3*g) - (b*c*(c^2*f^2 - g^2)*x)/g^3 + (b*c 
^3*f*x^2)/(4*g^2) - (b*c^3*x^3)/(9*g) - (a*(c*f - g)*(c*f + g)*Sqrt[1 - c^ 
2*x^2])/g^3 - (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/g^3 + 
(c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + ((1 - c^2*x^2)^( 
3/2)*(a + b*ArcCos[c*x]))/(3*g) - (c*f*(a + b*ArcCos[c*x])^2)/(4*b*g^2) + 
(c*(c^2*f^2 - g^2)*x*(a + b*ArcCos[c*x])^2)/(2*b*g^3) + ((c^2*f^2 - g^2)^2 
*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)) + ((c^2*f^2 - g^2)*(1 - c^2* 
x^2)*(a + b*ArcCos[c*x])^2)/(2*b*c*g^2*(f + g*x)) + (a*(c^2*f^2 - g^2)^(3/ 
2)*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/g^4 + (I 
*b*(c^2*f^2 - g^2)^(3/2)*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - 
Sqrt[c^2*f^2 - g^2])])/g^4 - (I*b*(c^2*f^2 - g^2)^(3/2)*ArcCos[c*x]*Log[1 
+ (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^4 + (b*(c^2*f^2 - 
g^2)^(3/2)*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])) 
])/g^4 - (b*(c^2*f^2 - g^2)^(3/2)*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f 
+ Sqrt[c^2*f^2 - g^2]))])/g^4))/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 5267

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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 1559, normalized size of antiderivative = 1.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1559\)
parts	\(\text {Expression too large to display}\)	\(1559\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x,method=_RETURNVERBOSE 
)
 

Output:

a/g*(1/3*(-(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*d*f/g*(-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)))-d*(c^2*f^2-g^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))))+b*(-1/4*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^ 
2-1)*arccos(c*x)^2*f*(2*c^2*f^2-3*g^2)*d*c/g^4-1/72*(-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)*(I+3*arccos(c*x))*d/(c^2*x^2-1)/g+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))*d*c/(c^2*x^2-1)/g^2-1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1 
)^(1/2)*c*x+c^2*x^2-1)*(4*I*c^2*f^2+4*arccos(c*x)*c^2*f^2-5*I*g^2-5*arccos 
(c*x)*g^2)*d/(c^2*x^2-1)/g^3-1/8*(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^( 
1/2)*x*c+c^2*x^2-1)*(4*arccos(c*x)*c^2*f^2-5*arccos(c*x)*g^2-4*I*c^2*f^2+5 
*I*g^2)*d/(c^2*x^2-1)/g^3+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1...
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F]

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

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))/(g*x+f),x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acos(c*x))/(f + g*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="max 
ima")
 

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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="gia 
c")
 

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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \] Input:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2))/(f + g*x),x)
 

Output:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(3/2))/(f + g*x), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\frac {\sqrt {d}\, d \left (-6 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3}+9 \mathit {asin} \left (c x \right ) a c f \,g^{2}+12 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}-12 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,g^{2}-6 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{2} x -2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{3} x^{2}+8 \sqrt {-c^{2} x^{2}+1}\, a \,g^{3}-6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}}{g x +f}d x \right ) b \,c^{2} g^{4}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g x +f}d x \right ) b \,g^{4}-2 a \,c^{2} f^{2} g \right )}{6 g^{4}} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))/(g*x+f),x)
 

Output:

(sqrt(d)*d*( - 6*asin(c*x)*a*c**3*f**3 + 9*asin(c*x)*a*c*f*g**2 + 12*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**2 - 12*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sq 
rt(c**2*f**2 - g**2))*a*g**2 - 6*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g + 3* 
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 + 8*sqrt( - c**2*x**2 + 1)*a*g**3 - 6*int((sqrt( - c**2*x**2 + 1) 
*acos(c*x)*x**2)/(f + g*x),x)*b*c**2*g**4 + 6*int((sqrt( - c**2*x**2 + 1)* 
acos(c*x))/(f + g*x),x)*b*g**4 - 2*a*c**2*f**2*g))/(6*g**4)