\(\int \frac {x^2 (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx\) [593]

Optimal result

Integrand size = 35, antiderivative size = 295 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {x (a+b \arccos (c x))^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 2.75 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.19 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {3 a^2 c \sqrt {d} e x+3 a^2 \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+3 a b \sqrt {d} e \left (2 c x \arccos (c x)+\sqrt {1-c^2 x^2} \arccos (c x)^2-2 \sqrt {1-c^2 x^2} \left (\log \left (\cos \left (\frac {1}{2} \arccos (c x)\right )\right )+\log \left (\sin \left (\frac {1}{2} \arccos (c x)\right )\right )\right )\right )+b^2 \sqrt {d} e \left (\arccos (c x) \left (3 c x \arccos (c x)+\sqrt {1-c^2 x^2} \left (3 i \arccos (c x)+\arccos (c x)^2-6 \left (\log \left (1-e^{i \arccos (c x)}\right )+\log \left (1+e^{i \arccos (c x)}\right )\right )\right )\right )+6 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+6 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{3 c^3 d^{3/2} e^2 \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:

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

Output:

(3*a^2*c*Sqrt[d]*e*x + 3*a^2*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTa 
n[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] 
+ 3*a*b*Sqrt[d]*e*(2*c*x*ArcCos[c*x] + Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2 - 2 
*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcCos[c*x]/2]] + Log[Sin[ArcCos[c*x]/2]])) + 
b^2*Sqrt[d]*e*(ArcCos[c*x]*(3*c*x*ArcCos[c*x] + Sqrt[1 - c^2*x^2]*((3*I)*A 
rcCos[c*x] + ArcCos[c*x]^2 - 6*(Log[1 - E^(I*ArcCos[c*x])] + Log[1 + E^(I* 
ArcCos[c*x])]))) + (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcCos[c*x])] 
+ (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcCos[c*x])]))/(3*c^3*d^(3/2)*e 
^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.57, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {5239, 5207, 5153, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5239

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(Sqrt[1 - c^2*x^2]*((x*(a + b*ArcCos[c*x])^2)/(c^2*Sqrt[1 - c^2*x^2]) + (a 
 + b*ArcCos[c*x])^3/(3*b*c^3) - (2*b*(((-1/2*I)*(a + b*ArcCos[c*x])^2)/b - 
 (2*I)*((I/2)*(a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])] + (b*Poly 
Log[2, E^((2*I)*ArcCos[c*x])])/4)))/c^3))/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c* 
e*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

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 5181

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

rule 5207

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

rule 5239

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[q])   Int[(h*x)^m*(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, h, m, n}, x] && EqQ[e*f + d*g, 0] & 
& EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (285 ) = 570\).

Time = 8.75 (sec) , antiderivative size = 711, normalized size of antiderivative = 2.41

method	result	size
default	\(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-\left (c x -1\right ) e d \left (c x +1\right )}}\right ) x^{2} c^{2} d e +\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-\left (c x -1\right ) e d \left (c x +1\right )}}\right ) d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \sqrt {c^{2} d e}\, \left (c x -1\right ) \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2}}+b^{2} \left (-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \arccos \left (c x \right )^{2}}{e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}+1\right )+i \arccos \left (c x \right ) \ln \left (1-\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, \sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )\right )}{e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arccos \left (c x \right ) c^{2} x^{2}+\arccos \left (c x \right )^{2} x^{2} c^{2}-2 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -2 i \arccos \left (c x \right )-\arccos \left (c x \right )^{2}+2 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )\right )}{\left (c^{2} x^{2}-1\right ) d^{2} e^{2} c^{3} \left (c x +1\right ) \left (c x -1\right )}\)	\(711\)
parts	\(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-\left (c x -1\right ) e d \left (c x +1\right )}}\right ) x^{2} c^{2} d e +\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-\left (c x -1\right ) e d \left (c x +1\right )}}\right ) d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \sqrt {c^{2} d e}\, \left (c x -1\right ) \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2}}+b^{2} \left (-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \arccos \left (c x \right )^{2}}{e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}+1\right )+i \arccos \left (c x \right ) \ln \left (1-\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -\sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, \sqrt {c x +i \sqrt {-c^{2} x^{2}+1}}\right )\right )}{e^{2} d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arccos \left (c x \right ) c^{2} x^{2}+\arccos \left (c x \right )^{2} x^{2} c^{2}-2 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{2} c^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -2 i \arccos \left (c x \right )-\arccos \left (c x \right )^{2}+2 \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )\right )}{\left (c^{2} x^{2}-1\right ) d^{2} e^{2} c^{3} \left (c x +1\right ) \left (c x -1\right )}\)	\(711\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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