\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arccos (c x))}{f+g x} \, dx\) [13]

Optimal result

Integrand size = 31, antiderivative size = 1598 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx =\text {Too large to display} \] Output:

1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/g+1/4*c*d^2*f*(c^2*f^2-2*g^2)*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/g^4/(-c^2*x^2+1)^(1/2)-1/2*c*d^2 
*(c^2*f^2-g^2)^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/g^5/(-c^2*x^ 
2+1)^(1/2)-1/2*d^2*(c^2*f^2-g^2)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^ 
2/b/c/g^6/(g*x+f)/(-c^2*x^2+1)^(1/2)-1/2*d^2*(c^2*f^2-g^2)^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^4/(g*x+f)-1/4*b*c^3*d 
^2*f*(c^2*f^2-2*g^2)*x^2*(-c^2*d*x^2+d)^(1/2)/g^4/(-c^2*x^2+1)^(1/2)-I*b*d 
^2*(c^2*f^2-g^2)^(5/2)*(-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^6/(-c^2*x^2+1)^(1/2)+I*b*d^2* 
(c^2*f^2-g^2)^(5/2)*(-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^6/(-c^2*x^2+1)^(1/2)-1/3*d*(c^2* 
f^2-2*g^2)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/g^3+b*d^2*(c^2*f^2-g^2)^ 
2*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/g^5+b*c*d^2*(c^2*f^2-g^2)^2*x*(-c^2*d*x 
^2+d)^(1/2)/g^5/(-c^2*x^2+1)^(1/2)-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(-c^2*d 
*x^2+d)^(1/2)*(a+b*arccos(c*x))/g^4+1/16*c*d^2*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)-1/3*b*c*d^2*(c^2*f^2-2*g^2)*x*(-c 
^2*d*x^2+d)^(1/2)/g^3/(-c^2*x^2+1)^(1/2)+1/16*b*c^3*d^2*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^2*(c^2*f^2-2*g^2)*x^3*(-c^2*d* 
x^2+d)^(1/2)/g^3/(-c^2*x^2+1)^(1/2)-1/16*b*c^5*d^2*f*x^4*(-c^2*d*x^2+d)^(1 
/2)/g^2/(-c^2*x^2+1)^(1/2)+a*d^2*(c^2*f^2-g^2)^2*(-c^2*d*x^2+d)^(1/2)/g...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6216\) vs. \(2(1598)=3196\).

Time = 16.71 (sec) , antiderivative size = 6216, normalized size of antiderivative = 3.89 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.10 (sec) , antiderivative size = 1008, 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, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx\)

\(\Big \downarrow \) 5277

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x}dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5267

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (\frac {b x^5 c^5}{25 g}-\frac {b f x^4 c^5}{16 g^2}-\frac {f x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) c^4}{4 g^2}+\frac {b \left (c^2 f^2-2 g^2\right ) x^3 c^3}{9 g^3}-\frac {b x^3 c^3}{45 g}-\frac {b f \left (c^2 f^2-2 g^2\right ) x^2 c^3}{4 g^4}+\frac {b f x^2 c^3}{16 g^2}-\frac {f \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} (a+b \arccos (c x)) c^2}{2 g^4}+\frac {f x \sqrt {1-c^2 x^2} (a+b \arccos (c x)) c^2}{8 g^2}+\frac {f \left (c^2 f^2-2 g^2\right ) (a+b \arccos (c x))^2 c}{4 b g^4}-\frac {\left (c^2 f^2-g^2\right )^2 x (a+b \arccos (c x))^2 c}{2 b g^5}+\frac {f (a+b \arccos (c x))^2 c}{16 b g^2}+\frac {b \left (c^2 f^2-g^2\right )^2 x c}{g^5}-\frac {b \left (c^2 f^2-2 g^2\right ) x c}{3 g^3}-\frac {2 b x c}{15 g}+\frac {b \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \arccos (c x)}{g^5}+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 g}-\frac {\left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 g^3}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 g}-\frac {a \left (c^2 f^2-g^2\right )^{5/2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6}-\frac {i b \left (c^2 f^2-g^2\right )^{5/2} \arccos (c x) \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}+\frac {i b \left (c^2 f^2-g^2\right )^{5/2} \arccos (c x) \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}-\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6}+\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6}+\frac {a \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2}}{g^5}-\frac {\left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b g^4 (f+g x) c}-\frac {\left (c^2 f^2-g^2\right )^3 (a+b \arccos (c x))^2}{2 b g^6 (f+g x) c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((-2*b*c*x)/(15*g) - (b*c*(c^2*f^2 - 2*g^2)*x)/(3 
*g^3) + (b*c*(c^2*f^2 - g^2)^2*x)/g^5 + (b*c^3*f*x^2)/(16*g^2) - (b*c^3*f* 
(c^2*f^2 - 2*g^2)*x^2)/(4*g^4) - (b*c^3*x^3)/(45*g) + (b*c^3*(c^2*f^2 - 2* 
g^2)*x^3)/(9*g^3) - (b*c^5*f*x^4)/(16*g^2) + (b*c^5*x^5)/(25*g) + (a*(c^2* 
f^2 - g^2)^2*Sqrt[1 - c^2*x^2])/g^5 + (b*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^ 
2]*ArcCos[c*x])/g^5 + (c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(8*g 
^2) - (c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2 
*g^4) - (c^4*f*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(4*g^2) - ((1 - 
c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(3*g) - ((c^2*f^2 - 2*g^2)*(1 - c^2*x^ 
2)^(3/2)*(a + b*ArcCos[c*x]))/(3*g^3) + ((1 - c^2*x^2)^(5/2)*(a + b*ArcCos 
[c*x]))/(5*g) + (c*f*(a + b*ArcCos[c*x])^2)/(16*b*g^2) + (c*f*(c^2*f^2 - 2 
*g^2)*(a + b*ArcCos[c*x])^2)/(4*b*g^4) - (c*(c^2*f^2 - g^2)^2*x*(a + b*Arc 
Cos[c*x])^2)/(2*b*g^5) - ((c^2*f^2 - g^2)^3*(a + b*ArcCos[c*x])^2)/(2*b*c* 
g^6*(f + g*x)) - ((c^2*f^2 - g^2)^2*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/( 
2*b*c*g^4*(f + g*x)) - (a*(c^2*f^2 - g^2)^(5/2)*ArcTan[(g + c^2*f*x)/(Sqrt 
[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/g^6 - (I*b*(c^2*f^2 - g^2)^(5/2)*ArcC 
os[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^6 + 
(I*b*(c^2*f^2 - g^2)^(5/2)*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f 
+ Sqrt[c^2*f^2 - g^2])])/g^6 - (b*(c^2*f^2 - g^2)^(5/2)*PolyLog[2, -((E^(I 
*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g^6 + (b*(c^2*f^2 - g^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 = 1.07 (sec) , antiderivative size = 2665, normalized size of antiderivative = 1.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2665\)
parts	\(\text {Expression too large to display}\)	\(2665\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x,method=_RETURNVERBOSE 
)
 

Output:

a/g*(1/5*(-(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)^(5/2)+ 
c^2*d*f/g*(-1/8*(-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)^(3/2)-3/16*(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*(-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*(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*(1/16*(-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*(8*c^4*f^4-20*c^2*f^2*g^2+15*g^4 
)*d^2*c/g^6+1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*I*(-...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{f + g x}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))/(g*x+f),x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acos(c*x))/(f + g*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \] Input:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x),x)
 

Output:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{f+g x} \, dx=\frac {\sqrt {d}\, d^{2} \left (120 \mathit {asin} \left (c x \right ) a \,c^{5} f^{5}-300 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3} g^{2}+225 \mathit {asin} \left (c x \right ) a c f \,g^{4}-240 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{4} f^{4}+480 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2} g^{2}-240 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,g^{4}+120 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{4} g -60 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{3} g^{2} x +40 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{2} g^{3} x^{2}-30 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,g^{4} x^{3}+24 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g^{5} x^{4}-280 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g^{3}+135 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{4} x -88 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{5} x^{2}+184 \sqrt {-c^{2} x^{2}+1}\, a \,g^{5}+120 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}}{g x +f}d x \right ) b \,c^{4} g^{6}-240 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}}{g x +f}d x \right ) b \,c^{2} g^{6}+120 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{g x +f}d x \right ) b \,g^{6}+72 a \,c^{4} f^{4} g -136 a \,c^{2} f^{2} g^{3}+40 a \,g^{5}\right )}{120 g^{6}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x))/(g*x+f),x)
 

Output:

(sqrt(d)*d**2*(120*asin(c*x)*a*c**5*f**5 - 300*asin(c*x)*a*c**3*f**3*g**2 
+ 225*asin(c*x)*a*c*f*g**4 - 240*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**4*f**4 + 480*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*g 
**2 - 240*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2 
*f**2 - g**2))*a*g**4 + 120*sqrt( - c**2*x**2 + 1)*a*c**4*f**4*g - 60*sqrt 
( - c**2*x**2 + 1)*a*c**4*f**3*g**2*x + 40*sqrt( - c**2*x**2 + 1)*a*c**4*f 
**2*g**3*x**2 - 30*sqrt( - c**2*x**2 + 1)*a*c**4*f*g**4*x**3 + 24*sqrt( - 
c**2*x**2 + 1)*a*c**4*g**5*x**4 - 280*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g 
**3 + 135*sqrt( - c**2*x**2 + 1)*a*c**2*f*g**4*x - 88*sqrt( - c**2*x**2 + 
1)*a*c**2*g**5*x**2 + 184*sqrt( - c**2*x**2 + 1)*a*g**5 + 120*int((sqrt( - 
 c**2*x**2 + 1)*acos(c*x)*x**4)/(f + g*x),x)*b*c**4*g**6 - 240*int((sqrt( 
- c**2*x**2 + 1)*acos(c*x)*x**2)/(f + g*x),x)*b*c**2*g**6 + 120*int((sqrt( 
 - c**2*x**2 + 1)*acos(c*x))/(f + g*x),x)*b*g**6 + 72*a*c**4*f**4*g - 136* 
a*c**2*f**2*g**3 + 40*a*g**5))/(120*g**6)