\(\int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx\) [552]

Optimal result

Integrand size = 32, antiderivative size = 714 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=\frac {2 a b e^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b^2 e^3 x \left (1-c^2 x^2\right )^{3/2} \arccos (c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 e^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 e^3 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i e^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {e^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {16 i b e^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 b e^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {8 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b^2 e^3 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \] Output:

2*a*b*e^3*x*(-c^2*x^2+1)^(3/2)/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+2*b^2*e^3* 
(-c^2*x^2+1)^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+2*b^2*e^3*x*(-c^2*x^2+1) 
^(3/2)*arccos(c*x)/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-4*e^3*(-c^2*x^2+1)*(a+ 
b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+4*e^3*x*(-c^2*x^2+1)*( 
a+b*arccos(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-4*I*e^3*(-c^2*x^2+1)^( 
3/2)*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-e^3*(-c^2*x^2+ 
1)^2*(a+b*arccos(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-e^3*(-c^2*x^2+ 
1)^(3/2)*(a+b*arccos(c*x))^3/b/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-16*I*b*e 
^3*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c 
/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+8*b*e^3*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c 
*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2) 
+8*I*b^2*e^3*(-c^2*x^2+1)^(3/2)*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c 
/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-8*I*b^2*e^3*(-c^2*x^2+1)^(3/2)*polylog(2 
,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-4*I*b^2* 
e^3*(-c^2*x^2+1)^(3/2)*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d 
)^(3/2)/(-c*e*x+e)^(3/2)
 

Mathematica [A] (verified)

Time = 11.14 (sec) , antiderivative size = 653, normalized size of antiderivative = 0.91 \[ \int \frac {(e-c e x)^{3/2} (a+b \arccos (c x))^2}{(d+c d x)^{3/2}} \, dx=\frac {-3 a^2 e (5+c x) \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}+9 a^2 \sqrt {d} e^{3/2} (1+c x) \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-3 a b e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \cot \left (\frac {1}{2} \arccos (c x)\right ) \left (4 \arccos (c x)-\cot \left (\frac {1}{2} \arccos (c x)\right ) \left (\arccos (c x)^2-8 \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right )-6 a b e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \cot \left (\frac {1}{2} \arccos (c x)\right ) \left (2 \arccos (c x)+\cot \left (\frac {1}{2} \arccos (c x)\right ) \left (c x+\sqrt {1-c^2 x^2} \arccos (c x)-\arccos (c x)^2+4 \log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right )+\frac {1}{2} b^2 e (1-c x) \sqrt {d+c d x} \sqrt {e-c e x} \cot \left (\frac {1}{2} \arccos (c x)\right ) \csc ^2\left (\frac {1}{2} \arccos (c x)\right ) \left (6-6 c^2 x^2+3 \left (-3+2 c x+c^2 x^2+2 i \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+2 \sqrt {1-c^2 x^2} \arccos (c x)^3-6 \sqrt {1-c^2 x^2} \arccos (c x) \left (c x+4 \log \left (1+e^{i \arccos (c x)}\right )\right )+24 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )+b^2 e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (\arccos (c x) \left (-6 \arccos (c x)+\cot \left (\frac {1}{2} \arccos (c x)\right ) \left (\arccos (c x) (6 i+\arccos (c x))-24 \log \left (1+e^{i \arccos (c x)}\right )\right )\right )+24 i \cot \left (\frac {1}{2} \arccos (c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )}{3 c d^2 (1+c x) \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-3*a^2*e*(5 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] + 9* 
a^2*Sqrt[d]*e^(3/2)*(1 + c*x)*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))] - 3*a*b*e*(1 - c*x)*S 
qrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(4*ArcCos[c*x] - Cot[Arc 
Cos[c*x]/2]*(ArcCos[c*x]^2 - 8*Log[Cos[ArcCos[c*x]/2]])) - 6*a*b*e*(1 - c* 
x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cot[ArcCos[c*x]/2]*(2*ArcCos[c*x] + Cot 
[ArcCos[c*x]/2]*(c*x + Sqrt[1 - c^2*x^2]*ArcCos[c*x] - ArcCos[c*x]^2 + 4*L 
og[Cos[ArcCos[c*x]/2]])) + (b^2*e*(1 - c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x 
]*Cot[ArcCos[c*x]/2]*Csc[ArcCos[c*x]/2]^2*(6 - 6*c^2*x^2 + 3*(-3 + 2*c*x + 
 c^2*x^2 + (2*I)*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + 2*Sqrt[1 - c^2*x^2]*Ar 
cCos[c*x]^3 - 6*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*(c*x + 4*Log[1 + E^(I*ArcCos 
[c*x])]) + (24*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcCos[c*x])]))/2 + b 
^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(ArcCos[c*x]*(-6*Ar 
cCos[c*x] + Cot[ArcCos[c*x]/2]*(ArcCos[c*x]*(6*I + ArcCos[c*x]) - 24*Log[1 
 + E^(I*ArcCos[c*x])])) + (24*I)*Cot[ArcCos[c*x]/2]*PolyLog[2, -E^(I*ArcCo 
s[c*x])]))/(3*c*d^2*(1 + c*x)*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 318, 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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5275

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (1-c^2 x^2\right )^{3/2} \left (-\frac {16 b \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c}+\frac {4 x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {4 (a+b \arccos (c x))^2}{c \sqrt {1-c^2 x^2}}+\frac {(a+b \arccos (c x))^3}{b c}+\frac {4 i (a+b \arccos (c x))^2}{c}-\frac {8 b \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{c}-2 a b x+\frac {8 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}-\frac {8 i b^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c}+\frac {4 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{c}-2 b^2 x \arccos (c x)+\frac {2 b^2 \sqrt {1-c^2 x^2}}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\)

Input:

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

Output:

(e^3*(1 - c^2*x^2)^(3/2)*(-2*a*b*x + (2*b^2*Sqrt[1 - c^2*x^2])/c - 2*b^2*x 
*ArcCos[c*x] + ((4*I)*(a + b*ArcCos[c*x])^2)/c - (4*(a + b*ArcCos[c*x])^2) 
/(c*Sqrt[1 - c^2*x^2]) + (4*x*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c^2*x^2] - ( 
Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/c + (a + b*ArcCos[c*x])^3/(b*c) - 
 (16*b*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/c - (8*b*(a + b*Arc 
Cos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])])/c + ((8*I)*b^2*PolyLog[2, -E^(I* 
ArcCos[c*x])])/c - ((8*I)*b^2*PolyLog[2, E^(I*ArcCos[c*x])])/c + ((4*I)*b^ 
2*PolyLog[2, E^((2*I)*ArcCos[c*x])])/c))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3 
/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 = 6.18 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (a +b \arccos \left (c x \right )\right )^{3} e}{\left (c x +1\right ) d^{2} \left (c x -1\right ) c b}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}-2 b^{2}+2 i \arccos \left (c x \right ) b^{2}+2 i a b \right ) e}{2 \left (c x +1\right ) d^{2} \left (c x -1\right ) c}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}-2 b^{2}-2 i \arccos \left (c x \right ) b^{2}-2 i a b \right ) e}{2 \left (c x +1\right ) d^{2} \left (c x -1\right ) c}-\frac {4 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x -1\right ) \left (\arccos \left (c x \right )^{2} b^{2}+2 \arccos \left (c x \right ) a b +a^{2}\right ) e}{\left (c x +1\right ) d^{2} \left (c x -1\right ) c}+\frac {8 \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, b \left (-i \arccos \left (c x \right )^{2} b +2 \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right ) \arccos \left (c x \right ) b -2 i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right ) b +2 a \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 a \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) e}{\left (c x +1\right ) d^{2} \left (c x -1\right ) c}\)	\(540\)

Input:

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

Output:

-(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c*x+1)/d^2/(c*x- 
1)/c*(a+b*arccos(c*x))^3*e/b-1/2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*( 
-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(arccos(c*x)^2*b^2+2*arccos(c*x)*a*b+a^2- 
2*b^2+2*I*arccos(c*x)*b^2+2*I*a*b)*e/(c*x+1)/d^2/(c*x-1)/c-1/2*(-e*(c*x-1) 
)^(1/2)*(d*(c*x+1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c* 
x)^2*b^2+2*arccos(c*x)*a*b+a^2-2*b^2-2*I*b^2*arccos(c*x)-2*I*a*b)*e/(c*x+1 
)/d^2/(c*x-1)/c-4*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-I*(-c^2*x^2+1)^(1 
/2)+c*x-1)*(arccos(c*x)^2*b^2+2*arccos(c*x)*a*b+a^2)*e/(c*x+1)/d^2/(c*x-1) 
/c+8*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2)*b*(-I*arccos( 
c*x)^2*b+2*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))*arccos(c*x)*b-2*I*polylog(2,-c*x 
-I*(-c^2*x^2+1)^(1/2))*b+2*a*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-2*a*ln(c*x+I*( 
-c^2*x^2+1)^(1/2)))*e/(c*x+1)/d^2/(c*x-1)/c
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*e*(6*sqrt(c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - sqrt( - 
c*x + 1)*a**2*c*x - 5*sqrt( - c*x + 1)*a**2 - 2*sqrt(c*x + 1)*int((sqrt( - 
 c*x + 1)*acos(c*x)*x)/(sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*a*b*c**2 + 2 
*sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x))/(sqrt(c*x + 1)*c*x + sqrt( 
c*x + 1)),x)*a*b*c - sqrt(c*x + 1)*int((sqrt( - c*x + 1)*acos(c*x)**2*x)/( 
sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c**2 + sqrt(c*x + 1)*int((sqrt( 
 - c*x + 1)*acos(c*x)**2)/(sqrt(c*x + 1)*c*x + sqrt(c*x + 1)),x)*b**2*c))/ 
(sqrt(d)*sqrt(c*x + 1)*c*d)