\(\int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx\) [213]

Optimal result

Integrand size = 29, antiderivative size = 303 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx=\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \arccos (c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \] Output:

1/64*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^2-1/32*b^2*x^3*(-c^2*d*x^2+d)^(1/2)-1/64 
*b^2*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/c^3/(-c^2*x^2+1)^(1/2)+1/8*b*x^2*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c/(-c^2*x^2+1)^(1/2)-1/8*b*c*x^4*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-1/8*x*(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccos(c*x))^2/c^2+1/4*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x 
))^2+1/24*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2 
)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.73 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx=\frac {96 a^2 c x \left (-1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}-96 a^2 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {12 a b \sqrt {d-c^2 d x^2} \left (-8 \arccos (c x)^2+\cos (4 \arccos (c x))+4 \arccos (c x) \sin (4 \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 \sqrt {d-c^2 d x^2} \left (32 \arccos (c x)^3-12 \arccos (c x) \cos (4 \arccos (c x))+\left (3-24 \arccos (c x)^2\right ) \sin (4 \arccos (c x))\right )}{\sqrt {1-c^2 x^2}}}{768 c^3} \] Input:

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

Output:

(96*a^2*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] - 96*a^2*Sqrt[d]*ArcTan[( 
c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (12*a*b*Sqrt[d - c^2* 
d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*ArcCos[c*x]] + 4*ArcCos[c*x]*Sin[4*ArcCos 
[c*x]]))/Sqrt[1 - c^2*x^2] - (b^2*Sqrt[d - c^2*d*x^2]*(32*ArcCos[c*x]^3 - 
12*ArcCos[c*x]*Cos[4*ArcCos[c*x]] + (3 - 24*ArcCos[c*x]^2)*Sin[4*ArcCos[c* 
x]]))/Sqrt[1 - c^2*x^2])/(768*c^3)
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5199, 5139, 262, 262, 223, 5211, 5139, 262, 223, 5153}

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.

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

Input:

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

Output:

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

Defintions of rubi rules used

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 5199

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

rule 5211

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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S 
imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.24

method	result	size
default	\(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(678\)
parts	\(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{24 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right )}{512 c^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{16 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-i+4 \arccos \left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(678\)

Input:

int(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/4*a^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+1 
/8*a^2/c^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^ 
2*(1/24*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arccos(c 
*x)^3+1/512*(-d*(c^2*x^2-1))^(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*arccos(c*x)+8*arccos(c*x)^2-1)/c^3/(c^2*x^2-1)+1/512*(-d*(c^2*x^2-1))^ 
(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^ 
2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(-4*I*arccos(c*x)+8*arccos(c* 
x)^2-1)/c^3/(c^2*x^2-1))+2*a*b*(1/16*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^( 
1/2)/c^3/(c^2*x^2-1)*arccos(c*x)^2+1/256*(-d*(c^2*x^2-1))^(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))/c^3/(c^2*x^2-1)+1/256*(-d*(c 
^2*x^2-1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+ 
1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(-I+4*arccos(c*x)) 
/c^3/(c^2*x^2-1))
 

Fricas [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2,x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

integrate(x**2*(-c**2*d*x**2+d)**(1/2)*(a+b*acos(c*x))**2,x)
 

Output:

Integral(x**2*sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))**2, x)
 

Maxima [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2,x, algorithm="maxim 
a")
 

Output:

1/8*a^2*(sqrt(-c^2*d*x^2 + d)*x/c^2 - 2*(-c^2*d*x^2 + d)^(3/2)*x/(c^2*d) + 
 sqrt(d)*arcsin(c*x)/c^3) + sqrt(d)*integrate((b^2*x^2*arctan2(sqrt(c*x + 
1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1) 
, c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.21 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx=\frac {1}{4} \, \sqrt {-c^{2} d x^{2} + d} a^{2} x^{3} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2} x}{8 \, c^{2}} - \frac {a^{2} d \log \left ({\left | -c \sqrt {-d} x + \sqrt {c^{2} x^{2} - 1} \sqrt {-d} \right |}\right )}{8 \, c^{3} \sqrt {-d}} + \frac {24 \, b^{2} c^{3} \sqrt {d} x^{4} \arccos \left (c x\right ) + 48 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{2} \sqrt {d} x^{3} \arccos \left (c x\right )^{2} + 24 \, a b c^{3} \sqrt {d} x^{4} + 96 \, \sqrt {-c^{2} x^{2} + 1} a b c^{2} \sqrt {d} x^{3} \arccos \left (c x\right ) - 6 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{2} \sqrt {d} x^{3} - 24 \, b^{2} c \sqrt {d} x^{2} \arccos \left (c x\right ) - 24 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \sqrt {d} x \arccos \left (c x\right )^{2} - 24 \, a b c \sqrt {d} x^{2} - 48 \, \sqrt {-c^{2} x^{2} + 1} a b \sqrt {d} x \arccos \left (c x\right ) - \frac {8 \, b^{2} \sqrt {d} \arccos \left (c x\right )^{3}}{c} + 3 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \sqrt {d} x - \frac {24 \, a b \sqrt {d} \arccos \left (c x\right )^{2}}{c} + \frac {3 \, b^{2} \sqrt {d} \arccos \left (c x\right )}{c} + \frac {3 \, a b \sqrt {d}}{c}}{192 \, c^{2}} \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2,x, algorithm="giac" 
)
 

Output:

1/4*sqrt(-c^2*d*x^2 + d)*a^2*x^3 - 1/8*sqrt(-c^2*d*x^2 + d)*a^2*x/c^2 - 1/ 
8*a^2*d*log(abs(-c*sqrt(-d)*x + sqrt(c^2*x^2 - 1)*sqrt(-d)))/(c^3*sqrt(-d) 
) + 1/192*(24*b^2*c^3*sqrt(d)*x^4*arccos(c*x) + 48*sqrt(-c^2*x^2 + 1)*b^2* 
c^2*sqrt(d)*x^3*arccos(c*x)^2 + 24*a*b*c^3*sqrt(d)*x^4 + 96*sqrt(-c^2*x^2 
+ 1)*a*b*c^2*sqrt(d)*x^3*arccos(c*x) - 6*sqrt(-c^2*x^2 + 1)*b^2*c^2*sqrt(d 
)*x^3 - 24*b^2*c*sqrt(d)*x^2*arccos(c*x) - 24*sqrt(-c^2*x^2 + 1)*b^2*sqrt( 
d)*x*arccos(c*x)^2 - 24*a*b*c*sqrt(d)*x^2 - 48*sqrt(-c^2*x^2 + 1)*a*b*sqrt 
(d)*x*arccos(c*x) - 8*b^2*sqrt(d)*arccos(c*x)^3/c + 3*sqrt(-c^2*x^2 + 1)*b 
^2*sqrt(d)*x - 24*a*b*sqrt(d)*arccos(c*x)^2/c + 3*b^2*sqrt(d)*arccos(c*x)/ 
c + 3*a*b*sqrt(d)/c)/c^2
 

Mupad [F(-1)]

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

int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(1/2),x)
 

Output:

int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

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

int(x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*acos(c*x))^2,x)
 

Output:

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