\(\int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx\) [553]

Optimal result

Integrand size = 32, antiderivative size = 544 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {8 i e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^3}{3 b c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {8 b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {8 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {4 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {2 e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} \arccos (c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac {32 b e^4 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) \log \left (1-i e^{i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac {32 i b^2 e^4 \left (1-c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}} \] Output:

8/3*I*e^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(5/2)/(-c*e*x 
+e)^(5/2)+1/3*e^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^3/b/c/(c*d*x+d)^(5/ 
2)/(-c*e*x+e)^(5/2)-8/3*b^2*e^4*(-c^2*x^2+1)^(5/2)*cot(1/4*Pi+1/2*arccos(c 
*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)+8/3*e^4*(-c^2*x^2+1)^(5/2)*(a+b*ar 
ccos(c*x))^2*cot(1/4*Pi+1/2*arccos(c*x))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2 
)-4/3*b*e^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))*csc(1/4*Pi+1/2*arccos(c*x 
))^2/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-2/3*e^4*(-c^2*x^2+1)^(5/2)*(a+b*ar 
ccos(c*x))^2*cot(1/4*Pi+1/2*arccos(c*x))*csc(1/4*Pi+1/2*arccos(c*x))^2/c/( 
c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)-32/3*b*e^4*(-c^2*x^2+1)^(5/2)*(a+b*arccos( 
c*x))*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2 
)+32/3*I*b^2*e^4*(-c^2*x^2+1)^(5/2)*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)) 
)/c/(c*d*x+d)^(5/2)/(-c*e*x+e)^(5/2)
 

Mathematica [A] (verified)

Time = 13.93 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.40 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((e - c*e*x)^(3/2)*(a + b*ArcCos[c*x])^2)/(d + c*d*x)^(5/2),x]
 

Output:

(32*a^2*e*(1 + 2*c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 
24*Sqrt[d]*e^(3/2)*(a + a*c*x)^2*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c* 
d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 16*a*b*e*(1 - c* 
x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(-((5 + 7*c*x)*ArcCo 
s[c*x]) + 3*ArcCos[c*x]^2*Cos[ArcCos[c*x]/2]^2*Cot[ArcCos[c*x]/2] - 2*Cot[ 
ArcCos[c*x]/2]*(1 + 7*(1 + c*x)*Log[Cos[ArcCos[c*x]/2]])) - 16*b^2*e*(1 - 
c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(ArcCos[c*x]*(ArcC 
os[c*x] - 2*Cot[ArcCos[c*x]/2]) + Cos[ArcCos[c*x]/2]^2*(4 - ArcCos[c*x]^2 
+ I*ArcCos[c*x]*Cot[ArcCos[c*x]/2]*(ArcCos[c*x] + (4*I)*Log[1 + E^(I*ArcCo 
s[c*x])])) + (4*I)*Cos[ArcCos[c*x]/2]^2*Cot[ArcCos[c*x]/2]*PolyLog[2, -E^( 
I*ArcCos[c*x])]) - b^2*e*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 
- c^2*x^2]*Csc[ArcCos[c*x]/2]^2*(8*ArcCos[c*x]^2 - 2*Sqrt[1 - c^2*x^2]*(-4 
*Sqrt[1 - c^2*x^2] + 4*ArcCos[c*x] + 7*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2)*Cs 
c[ArcCos[c*x]/2]^2 + (1 - c^2*x^2)^(3/2)*ArcCos[c*x]*Csc[ArcCos[c*x]/2]^4* 
(ArcCos[c*x]*(7*I + ArcCos[c*x]) - 28*Log[1 + E^(I*ArcCos[c*x])]) + (28*I) 
*(1 - c^2*x^2)^(3/2)*Csc[ArcCos[c*x]/2]^4*PolyLog[2, -E^(I*ArcCos[c*x])]) 
- 32*a*b*e*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(- 
(Cot[ArcCos[c*x]/2]*(1 + (1 + c*x)*Log[Cos[ArcCos[c*x]/2]])) + ArcCos[c*x] 
*Sin[ArcCos[c*x]/2]^2))/(24*c*d^3*(1 + c*x)^2*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5179, 27, 5275, 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.

\(\displaystyle \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(c d x+d)^{5/2}} \, dx\)

\(\Big \downarrow \) 5179

\(\displaystyle \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {e^4 (1-c x)^4 (a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^4 \left (1-c^2 x^2\right )^{5/2} \int \frac {(1-c x)^4 (a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 5275

\(\displaystyle \frac {e^4 \left (1-c^2 x^2\right )^{5/2} \int \left (-\frac {4 (a+b \arccos (c x))^2}{(c x+1) \sqrt {1-c^2 x^2}}+\frac {4 (a+b \arccos (c x))^2}{(c x+1)^2 \sqrt {1-c^2 x^2}}+\frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}\right )dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^4 \left (1-c^2 x^2\right )^{5/2} \left (-\frac {(a+b \arccos (c x))^3}{3 b c}-\frac {8 i (a+b \arccos (c x))^2}{3 c}+\frac {32 b \log \left (1+e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{3 c}+\frac {8 \tan \left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))^2}{3 c}+\frac {4 b \sec ^2\left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))}{3 c}-\frac {2 \tan \left (\frac {1}{2} \arccos (c x)\right ) \sec ^2\left (\frac {1}{2} \arccos (c x)\right ) (a+b \arccos (c x))^2}{3 c}-\frac {32 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{3 c}-\frac {8 b^2 \tan \left (\frac {1}{2} \arccos (c x)\right )}{3 c}\right )}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

Input:

Int[((e - c*e*x)^(3/2)*(a + b*ArcCos[c*x])^2)/(d + c*d*x)^(5/2),x]
 

Output:

(e^4*(1 - c^2*x^2)^(5/2)*((((-8*I)/3)*(a + b*ArcCos[c*x])^2)/c - (a + b*Ar 
cCos[c*x])^3/(3*b*c) + (32*b*(a + b*ArcCos[c*x])*Log[1 + E^(I*ArcCos[c*x]) 
])/(3*c) - (((32*I)/3)*b^2*PolyLog[2, -E^(I*ArcCos[c*x])])/c + (4*b*(a + b 
*ArcCos[c*x])*Sec[ArcCos[c*x]/2]^2)/(3*c) - (8*b^2*Tan[ArcCos[c*x]/2])/(3* 
c) + (8*(a + b*ArcCos[c*x])^2*Tan[ArcCos[c*x]/2])/(3*c) - (2*(a + b*ArcCos 
[c*x])^2*Sec[ArcCos[c*x]/2]^2*Tan[ArcCos[c*x]/2])/(3*c)))/((d + c*d*x)^(5/ 
2)*(e - c*e*x)^(5/2))
 

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 5179

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

rule 5275

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x] 
)^n/Sqrt[d + e*x^2], (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] && ILtQ[p + 1/2, 
 0] && GtQ[d, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (a +b \arccos \left (c x \right )\right )^{3} e}{3 \left (c x +1\right ) d^{3} \left (c x -1\right ) c b}+\frac {4 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-2 i \sqrt {-c^{2} x^{2}+1}-c x -1\right ) \left (12 b^{2} c^{2} x^{2} \arccos \left (c x \right )^{2}-8 i a b c x +24 a b \,c^{2} x^{2} \arccos \left (c x \right )+2 i \sqrt {-c^{2} x^{2}+1}\, b^{2} c x +15 \arccos \left (c x \right )^{2} b^{2} c x -4 i a b +4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) b^{2} c x -4 i \arccos \left (c x \right ) b^{2}+12 a^{2} c^{2} x^{2}-10 x^{2} c^{2} b^{2}+30 \arccos \left (c x \right ) a b c x -8 i \arccos \left (c x \right ) b^{2} c x +4 \sqrt {-c^{2} x^{2}+1}\, a b c x +5 \arccos \left (c x \right )^{2} b^{2}-4 i \arccos \left (c x \right ) b^{2} c^{2} x^{2}+2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, b^{2}-4 i a b \,c^{2} x^{2}+15 a^{2} c x -16 c x \,b^{2}+10 \arccos \left (c x \right ) a b +2 i \sqrt {-c^{2} x^{2}+1}\, b^{2}+2 \sqrt {-c^{2} x^{2}+1}\, a b +5 a^{2}-6 b^{2}\right ) e}{3 \left (12 c^{4} x^{4}+39 c^{3} x^{3}+47 c^{2} x^{2}+25 c x +5\right ) d^{3} \left (c x -1\right ) c}+\frac {16 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, b \left (\arccos \left (c x \right )^{2} b +2 i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) b +2 \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right ) b +2 i \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) a -2 i \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) a \right ) e}{3 \left (c x +1\right ) d^{3} \left (c x -1\right ) c}\)	\(647\)

Input:

int((-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x,method=_RETURNV 
ERBOSE)
 

Output:

1/3*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x+1)/d^3/(c 
*x-1)/c*(a+b*arccos(c*x))^3*e/b+4/3*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*( 
-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-2*I*(-c^2*x^2+1)^(1/2)-c*x-1)*(12*b^ 
2*c^2*x^2*arccos(c*x)^2-8*I*a*b*c*x+24*a*b*c^2*x^2*arccos(c*x)+2*I*(-c^2*x 
^2+1)^(1/2)*b^2*c*x+15*arccos(c*x)^2*b^2*c*x-4*I*a*b+4*(-c^2*x^2+1)^(1/2)* 
arccos(c*x)*b^2*c*x-4*I*arccos(c*x)*b^2+12*a^2*c^2*x^2-10*x^2*c^2*b^2+30*a 
rccos(c*x)*a*b*c*x-8*I*arccos(c*x)*b^2*c*x+4*(-c^2*x^2+1)^(1/2)*a*b*c*x+5* 
arccos(c*x)^2*b^2-4*I*arccos(c*x)*b^2*c^2*x^2+2*arccos(c*x)*(-c^2*x^2+1)^( 
1/2)*b^2-4*I*a*b*c^2*x^2+15*a^2*c*x-16*c*x*b^2+10*arccos(c*x)*a*b+2*I*(-c^ 
2*x^2+1)^(1/2)*b^2+2*(-c^2*x^2+1)^(1/2)*a*b+5*a^2-6*b^2)*e/(12*c^4*x^4+39* 
c^3*x^3+47*c^2*x^2+25*c*x+5)/d^3/(c*x-1)/c+16/3*I*(-c^2*x^2+1)^(1/2)*(d*(c 
*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(c*x+1)/d^3/(c*x-1)/c*b*(arccos(c*x)^2*b+2 
*I*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))*b+2*polylog(2,-c*x-I*(-c^2*x 
^2+1)^(1/2))*b+2*I*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))*a-2*I*ln(c*x+I*(-c^2*x^2 
+1)^(1/2))*a)*e
 

Fricas [F]

\[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x, algorith 
m="fricas")
 

Output:

integral(-(a^2*c*e*x - a^2*e + (b^2*c*e*x - b^2*e)*arccos(c*x)^2 + 2*(a*b* 
c*e*x - a*b*e)*arccos(c*x))*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^3*d^3*x^3 
+ 3*c^2*d^3*x^2 + 3*c*d^3*x + d^3), x)
 

Sympy [F]

\[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\int \frac {\left (- e \left (c x - 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((-c*e*x+e)**(3/2)*(a+b*acos(c*x))**2/(c*d*x+d)**(5/2),x)
 

Output:

Integral((-e*(c*x - 1))**(3/2)*(a + b*acos(c*x))**2/(d*(c*x + 1))**(5/2), 
x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x, algorith 
m="maxima")
 

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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\int { \frac {{\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/(c*d*x+d)^(5/2),x, algorith 
m="giac")
 

Output:

integrate((-c*e*x + e)^(3/2)*(b*arccos(c*x) + a)^2/(c*d*x + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{3/2}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \] Input:

int(((a + b*acos(c*x))^2*(e - c*e*x)^(3/2))/(d + c*d*x)^(5/2),x)
 

Output:

int(((a + b*acos(c*x))^2*(e - c*e*x)^(3/2))/(d + c*d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{5/2}} \, dx=\frac {\sqrt {e}\, e \left (-6 \sqrt {c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2} c x -6 \sqrt {c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+8 \sqrt {-c x +1}\, a^{2} c x +4 \sqrt {-c x +1}\, a^{2}-6 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b \,c^{3} x -6 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b \,c^{2}+6 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b \,c^{2} x +6 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) a b c -3 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c^{3} x -3 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c^{2}+3 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2}}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c^{2} x +3 \sqrt {c x +1}\, \left (\int \frac {\sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2}}{\sqrt {c x +1}\, c^{2} x^{2}+2 \sqrt {c x +1}\, c x +\sqrt {c x +1}}d x \right ) b^{2} c \right )}{3 \sqrt {d}\, \sqrt {c x +1}\, c \,d^{2} \left (c x +1\right )} \] Input:

int((-c*e*x+e)^(3/2)*(a+b*acos(c*x))^2/(c*d*x+d)^(5/2),x)
 

Output:

(sqrt(e)*e*( - 6*sqrt(c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2*c*x - 6 
*sqrt(c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 8*sqrt( - c*x + 1)*a* 
*2*c*x + 4*sqrt( - c*x + 1)*a**2 - 6*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*a 
cos(c*x)*x)/(sqrt(c*x + 1)*c**2*x**2 + 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1) 
),x)*a*b*c**3*x - 6*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)*x)/(sqrt 
(c*x + 1)*c**2*x**2 + 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*a*b*c**2 + 6 
*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x))/(sqrt(c*x + 1)*c**2*x**2 + 
 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*a*b*c**2*x + 6*sqrt(c*x + 1)*int( 
(sqrt( - c*x + 1)*acos(c*x))/(sqrt(c*x + 1)*c**2*x**2 + 2*sqrt(c*x + 1)*c* 
x + sqrt(c*x + 1)),x)*a*b*c - 3*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c 
*x)**2*x)/(sqrt(c*x + 1)*c**2*x**2 + 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1)), 
x)*b**2*c**3*x - 3*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)**2*x)/(sq 
rt(c*x + 1)*c**2*x**2 + 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c**2 
+ 3*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)**2)/(sqrt(c*x + 1)*c**2* 
x**2 + 2*sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c**2*x + 3*sqrt(c*x + 
1)*int((sqrt( - c*x + 1)*acos(c*x)**2)/(sqrt(c*x + 1)*c**2*x**2 + 2*sqrt(c 
*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c))/(3*sqrt(d)*sqrt(c*x + 1)*c*d**2*( 
c*x + 1))