\(\int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2 \, dx\) [549]

Optimal result

Integrand size = 32, antiderivative size = 362 \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2 \, dx=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \arccos (c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}} \] Output:

-1/32*b^2*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)-15*b^2*x*(c*d*x+d)^(3/2)*(-c* 
e*x+e)^(3/2)/(-64*c^2*x^2+64)+9/64*b^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*ar 
ccos(c*x)/c/(-c^2*x^2+1)^(3/2)-3/8*b*c*x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2 
)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(3/2)+1/8*b*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3 
/2)*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c+1/4*x*(c*d*x+d)^(3/2)*(-c*e*x+e 
)^(3/2)*(a+b*arccos(c*x))^2+3*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcc 
os(c*x))^2/(-8*c^2*x^2+8)+1/8*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arccos 
(c*x))^3/b/c/(-c^2*x^2+1)^(3/2)
 

Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.03 \[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {-32 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^3-96 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-8 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^2 (12 a-8 b \sin (2 \arccos (c x))+b \sin (4 \arccos (c x)))+d e \sqrt {d+c d x} \sqrt {e-c e x} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+64 a b \cos (2 \arccos (c x))-4 a b \cos (4 \arccos (c x))-32 b^2 \sin (2 \arccos (c x))+b^2 \sin (4 \arccos (c x))\right )-4 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x) (-16 b \cos (2 \arccos (c x))+b \cos (4 \arccos (c x))+4 a (-8 \sin (2 \arccos (c x))+\sin (4 \arccos (c x))))}{256 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-32*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcCos[c*x]^3 - 96*a^2*d^(3/2 
)*e^(3/2)*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))] - 8*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] 
*ArcCos[c*x]^2*(12*a - 8*b*Sin[2*ArcCos[c*x]] + b*Sin[4*ArcCos[c*x]]) + d* 
e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2* 
c^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcCos[c*x]] - 4*a*b*Cos[4*ArcCos 
[c*x]] - 32*b^2*Sin[2*ArcCos[c*x]] + b^2*Sin[4*ArcCos[c*x]]) - 4*b*d*e*Sqr 
t[d + c*d*x]*Sqrt[e - c*e*x]*ArcCos[c*x]*(-16*b*Cos[2*ArcCos[c*x]] + b*Cos 
[4*ArcCos[c*x]] + 4*a*(-8*Sin[2*ArcCos[c*x]] + Sin[4*ArcCos[c*x]])))/(256* 
c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.76, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {5179, 5159, 5157, 5139, 262, 223, 5153, 5183, 211, 211, 223}

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.

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

\(\Big \downarrow \) 5179

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx+\frac {3}{4} \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{2} b c \left (-\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )+\frac {3}{4} \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{2} b c \left (-\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )+\frac {3}{4} \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{2} b c \left (-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )+\frac {3}{4} \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {1}{2} b c \left (-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}\right )+\frac {3}{4} \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

Input:

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

Output:

((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*((x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos 
[c*x])^2)/4 + (3*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/2 - (a + b*A 
rcCos[c*x])^3/(6*b*c) + b*c*((x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/2*(x*S 
qrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)))/4 + (b*c*(-1/4*((1 - c^ 
2*x^2)^2*(a + b*ArcCos[c*x]))/c^2 - (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x 
*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/2))/(1 - c^2*x^2)^ 
(3/2)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5157

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcCos[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], 
x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

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 5183

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 
1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.23 (sec) , antiderivative size = 1099, normalized size of antiderivative = 3.04

method	result	size
default	\(\text {Expression too large to display}\)	\(1099\)
parts	\(\text {Expression too large to display}\)	\(1099\)

Input:

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

Output:

-1/4*a^2/c/e*(c*d*x+d)^(3/2)*(-c*e*x+e)^(5/2)-1/4*a^2*d/c/e*(c*d*x+d)^(1/2 
)*(-c*e*x+e)^(5/2)+1/8*a^2*d/c*(-c*e*x+e)^(3/2)*(c*d*x+d)^(1/2)+3/8*a^2*d* 
e/c*(-c*e*x+e)^(1/2)*(c*d*x+d)^(1/2)+3/8*a^2*d^2*e^2*((-c*e*x+e)*(c*d*x+d) 
)^(1/2)/(-c*e*x+e)^(1/2)/(c*d*x+d)^(1/2)/(c^2*d*e)^(1/2)*arctan((c^2*d*e)^ 
(1/2)*x/(-c^2*d*e*x^2+d*e)^(1/2))+b^2*(1/8*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^ 
(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/c*arccos(c*x)^3*d*e-1/512*(d*(c*x+1)) 
^(1/2)*(-e*(c*x-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4 
*c^4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(4*I*arcco 
s(c*x)+8*arccos(c*x)^2-1)*d*e/(c^2*x^2-1)/c+1/16*(d*(c*x+1))^(1/2)*(-e*(c* 
x-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2 
)-2*c*x)*(2*arccos(c*x)^2-1-2*I*arccos(c*x))*d*e/(c^2*x^2-1)/c-3/512*(d*(c 
*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(20* 
I*arccos(c*x)+24*arccos(c*x)^2-11)*cos(3*arccos(c*x))*d*e/(c^2*x^2-1)/c-1/ 
512*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2) 
-I)*(68*I*arccos(c*x)+56*arccos(c*x)^2-31)*sin(3*arccos(c*x))*d*e/(c^2*x^2 
-1)/c)+2*a*b*(3/16*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-c^2*x^2+1)^(1/2) 
/(c^2*x^2-1)/c*arccos(c*x)^2*d*e-1/256*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2 
)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+4*c*x-8*I*(-c^2*x^2 
+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(I+4*arccos(c*x))*d*e/(c^2*x^2-1)/ 
c+1/16*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d e \left (-6 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}+5 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x -16 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+16 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )d x \right ) a b c -8 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+8 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{8 c} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*d*e*( - 6*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - 2*sqrt(c* 
x + 1)*sqrt( - c*x + 1)*a**2*c**3*x**3 + 5*sqrt(c*x + 1)*sqrt( - c*x + 1)* 
a**2*c*x - 16*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)*x**2,x)*a*b*c** 
3 + 16*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x),x)*a*b*c - 8*int(sqrt( 
c*x + 1)*sqrt( - c*x + 1)*acos(c*x)**2*x**2,x)*b**2*c**3 + 8*int(sqrt(c*x 
+ 1)*sqrt( - c*x + 1)*acos(c*x)**2,x)*b**2*c))/(8*c)