\(\int x (d-c^2 d x^2)^2 (a+b \arccos (c x))^2 \, dx\) [170]

Optimal result

Integrand size = 25, antiderivative size = 218 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=-\frac {5}{96} b^2 d^2 x^2+\frac {5 b^2 d^2 \left (1-c^2 x^2\right )^2}{288 c^2}+\frac {b^2 d^2 \left (1-c^2 x^2\right )^3}{108 c^2}+\frac {5 b d^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{48 c}+\frac {5 b d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{72 c}+\frac {b d^2 x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{18 c}+\frac {5 d^2 (a+b \arccos (c x))^2}{96 c^2}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2} \] Output:

-5/96*b^2*d^2*x^2+5/288*b^2*d^2*(-c^2*x^2+1)^2/c^2+1/108*b^2*d^2*(-c^2*x^2 
+1)^3/c^2+5/48*b*d^2*x*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c+5/72*b*d^2*x 
*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/c+1/18*b*d^2*x*(-c^2*x^2+1)^(5/2)*(a 
+b*arccos(c*x))/c+5/96*d^2*(a+b*arccos(c*x))^2/c^2-1/6*d^2*(-c^2*x^2+1)^3* 
(a+b*arccos(c*x))^2/c^2
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.96 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^2 \left (c x \left (b^2 c x \left (-99+39 c^2 x^2-8 c^4 x^4\right )+144 a^2 c x \left (3-3 c^2 x^2+c^4 x^4\right )-6 a b \sqrt {1-c^2 x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )\right )+6 b c x \left (b \sqrt {1-c^2 x^2} \left (-33+26 c^2 x^2-8 c^4 x^4\right )+48 a c x \left (3-3 c^2 x^2+c^4 x^4\right )\right ) \arccos (c x)+9 b^2 \left (-11+48 c^2 x^2-48 c^4 x^4+16 c^6 x^6\right ) \arccos (c x)^2+198 a b \arcsin (c x)\right )}{864 c^2} \] Input:

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

Output:

(d^2*(c*x*(b^2*c*x*(-99 + 39*c^2*x^2 - 8*c^4*x^4) + 144*a^2*c*x*(3 - 3*c^2 
*x^2 + c^4*x^4) - 6*a*b*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + 
 6*b*c*x*(b*Sqrt[1 - c^2*x^2]*(-33 + 26*c^2*x^2 - 8*c^4*x^4) + 48*a*c*x*(3 
 - 3*c^2*x^2 + c^4*x^4))*ArcCos[c*x] + 9*b^2*(-11 + 48*c^2*x^2 - 48*c^4*x^ 
4 + 16*c^6*x^6)*ArcCos[c*x]^2 + 198*a*b*ArcSin[c*x]))/(864*c^2)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5183, 5159, 241, 5159, 244, 2009, 5157, 15, 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.

\(\displaystyle \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx\)

\(\Big \downarrow \) 5183

\(\displaystyle -\frac {b d^2 \int \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 5159

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 5159

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 244

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 5157

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {b d^2 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}-\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}\)

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {d^2 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{6 c^2}-\frac {b d^2 \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )}{3 c}\)

Input:

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

Output:

-1/6*(d^2*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^2)/c^2 - (b*d^2*(-1/36*(b*(1 
 - c^2*x^2)^3)/c + (x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/6 + (5*((b* 
c*(x^2/2 - (c^2*x^4)/4))/4 + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/4 
 + (3*((b*c*x^2)/4 + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (a + b* 
ArcCos[c*x])^2/(4*b*c)))/4))/6))/(3*c)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 241

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

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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 [A] (verified)

Time = 0.27 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {5 \arccos \left (c x \right )^{2}}{96}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{2}+\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {\arccos \left (c x \right )}{6}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}-\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}-\frac {5 \arcsin \left (c x \right )}{96}\right )}{c^{2}}\)	\(265\)
default	\(\frac {\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {5 \arccos \left (c x \right )^{2}}{96}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{2}+\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {\arccos \left (c x \right )}{6}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}-\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}-\frac {5 \arcsin \left (c x \right )}{96}\right )}{c^{2}}\)	\(265\)
parts	\(\frac {d^{2} a^{2} \left (c^{2} x^{2}-1\right )^{3}}{6 c^{2}}+\frac {d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {5 \arccos \left (c x \right )^{2}}{96}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )}{c^{2}}+\frac {2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{2}+\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {\arccos \left (c x \right )}{6}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {13 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{144}-\frac {11 c x \sqrt {-c^{2} x^{2}+1}}{96}-\frac {5 \arcsin \left (c x \right )}{96}\right )}{c^{2}}\)	\(270\)
orering	\(\frac {\left (728 c^{8} x^{8}-3251 c^{6} x^{6}+6466 c^{4} x^{4}-3177 c^{2} x^{2}+594\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}}{1728 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (120 c^{6} x^{6}-571 c^{4} x^{4}+1323 c^{2} x^{2}-396\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-4 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{1728 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}+\frac {x \left (8 c^{4} x^{4}-39 c^{2} x^{2}+99\right ) \left (-12 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2} x -\frac {4 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+8 x^{3} d^{2} c^{4} \left (a +b \arccos \left (c x \right )\right )^{2}+\frac {16 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{1728 c^{2} \left (c x -1\right ) \left (c x +1\right )}\)	\(480\)

Input:

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

Output:

1/c^2*(1/6*d^2*a^2*(c^2*x^2-1)^3+d^2*b^2*(1/6*arccos(c*x)^2*(c^2*x^2-1)^3+ 
1/144*arccos(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)+26*c^3*x^3*(-c^2*x^2+1)^( 
1/2)-33*c*x*(-c^2*x^2+1)^(1/2)+15*arccos(c*x))-5/96*arccos(c*x)^2-1/108*c^ 
6*x^6+13/288*c^4*x^4-11/96*c^2*x^2)+2*d^2*a*b*(1/6*arccos(c*x)*c^6*x^6-1/2 
*c^4*x^4*arccos(c*x)+1/2*c^2*x^2*arccos(c*x)-1/6*arccos(c*x)-1/36*c^5*x^5* 
(-c^2*x^2+1)^(1/2)+13/144*c^3*x^3*(-c^2*x^2+1)^(1/2)-11/96*c*x*(-c^2*x^2+1 
)^(1/2)-5/96*arcsin(c*x)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.28 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {8 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d^{2} x^{6} - 3 \, {\left (144 \, a^{2} - 13 \, b^{2}\right )} c^{4} d^{2} x^{4} + 9 \, {\left (48 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} x^{2} + 9 \, {\left (16 \, b^{2} c^{6} d^{2} x^{6} - 48 \, b^{2} c^{4} d^{2} x^{4} + 48 \, b^{2} c^{2} d^{2} x^{2} - 11 \, b^{2} d^{2}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (16 \, a b c^{6} d^{2} x^{6} - 48 \, a b c^{4} d^{2} x^{4} + 48 \, a b c^{2} d^{2} x^{2} - 11 \, a b d^{2}\right )} \arccos \left (c x\right ) - 6 \, {\left (8 \, a b c^{5} d^{2} x^{5} - 26 \, a b c^{3} d^{2} x^{3} + 33 \, a b c d^{2} x + {\left (8 \, b^{2} c^{5} d^{2} x^{5} - 26 \, b^{2} c^{3} d^{2} x^{3} + 33 \, b^{2} c d^{2} x\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{864 \, c^{2}} \] Input:

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

Output:

1/864*(8*(18*a^2 - b^2)*c^6*d^2*x^6 - 3*(144*a^2 - 13*b^2)*c^4*d^2*x^4 + 9 
*(48*a^2 - 11*b^2)*c^2*d^2*x^2 + 9*(16*b^2*c^6*d^2*x^6 - 48*b^2*c^4*d^2*x^ 
4 + 48*b^2*c^2*d^2*x^2 - 11*b^2*d^2)*arccos(c*x)^2 + 18*(16*a*b*c^6*d^2*x^ 
6 - 48*a*b*c^4*d^2*x^4 + 48*a*b*c^2*d^2*x^2 - 11*a*b*d^2)*arccos(c*x) - 6* 
(8*a*b*c^5*d^2*x^5 - 26*a*b*c^3*d^2*x^3 + 33*a*b*c*d^2*x + (8*b^2*c^5*d^2* 
x^5 - 26*b^2*c^3*d^2*x^3 + 33*b^2*c*d^2*x)*arccos(c*x))*sqrt(-c^2*x^2 + 1) 
)/c^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (202) = 404\).

Time = 0.68 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.00 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{6}}{6} - \frac {a^{2} c^{2} d^{2} x^{4}}{2} + \frac {a^{2} d^{2} x^{2}}{2} + \frac {a b c^{4} d^{2} x^{6} \operatorname {acos}{\left (c x \right )}}{3} - \frac {a b c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} - a b c^{2} d^{2} x^{4} \operatorname {acos}{\left (c x \right )} + \frac {13 a b c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{72} + a b d^{2} x^{2} \operatorname {acos}{\left (c x \right )} - \frac {11 a b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{48 c} - \frac {11 a b d^{2} \operatorname {acos}{\left (c x \right )}}{48 c^{2}} + \frac {b^{2} c^{4} d^{2} x^{6} \operatorname {acos}^{2}{\left (c x \right )}}{6} - \frac {b^{2} c^{4} d^{2} x^{6}}{108} - \frac {b^{2} c^{3} d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{18} - \frac {b^{2} c^{2} d^{2} x^{4} \operatorname {acos}^{2}{\left (c x \right )}}{2} + \frac {13 b^{2} c^{2} d^{2} x^{4}}{288} + \frac {13 b^{2} c d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{72} + \frac {b^{2} d^{2} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {11 b^{2} d^{2} x^{2}}{96} - \frac {11 b^{2} d^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{48 c} - \frac {11 b^{2} d^{2} \operatorname {acos}^{2}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{2} \left (a + \frac {\pi b}{2}\right )^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c**4*d**2*x**6/6 - a**2*c**2*d**2*x**4/2 + a**2*d**2*x**2/ 
2 + a*b*c**4*d**2*x**6*acos(c*x)/3 - a*b*c**3*d**2*x**5*sqrt(-c**2*x**2 + 
1)/18 - a*b*c**2*d**2*x**4*acos(c*x) + 13*a*b*c*d**2*x**3*sqrt(-c**2*x**2 
+ 1)/72 + a*b*d**2*x**2*acos(c*x) - 11*a*b*d**2*x*sqrt(-c**2*x**2 + 1)/(48 
*c) - 11*a*b*d**2*acos(c*x)/(48*c**2) + b**2*c**4*d**2*x**6*acos(c*x)**2/6 
 - b**2*c**4*d**2*x**6/108 - b**2*c**3*d**2*x**5*sqrt(-c**2*x**2 + 1)*acos 
(c*x)/18 - b**2*c**2*d**2*x**4*acos(c*x)**2/2 + 13*b**2*c**2*d**2*x**4/288 
 + 13*b**2*c*d**2*x**3*sqrt(-c**2*x**2 + 1)*acos(c*x)/72 + b**2*d**2*x**2* 
acos(c*x)**2/2 - 11*b**2*d**2*x**2/96 - 11*b**2*d**2*x*sqrt(-c**2*x**2 + 1 
)*acos(c*x)/(48*c) - 11*b**2*d**2*acos(c*x)**2/(96*c**2), Ne(c, 0)), (d**2 
*x**2*(a + pi*b/2)**2/2, True))
 

Maxima [F]

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

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

Output:

1/6*a^2*c^4*d^2*x^6 - 1/2*a^2*c^2*d^2*x^4 + 1/144*(48*x^6*arccos(c*x) - (8 
*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2 
*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^4*d^2 - 1/8*(8*x^4*arccos(c 
*x) - (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsi 
n(c*x)/c^5)*c)*a*b*c^2*d^2 + 1/2*a^2*d^2*x^2 + 1/2*(2*x^2*arccos(c*x) - c* 
(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d^2 + 1/6*(b^2*c^4*d^2*x 
^6 - 3*b^2*c^2*d^2*x^4 + 3*b^2*d^2*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 
1), c*x)^2 - integrate(1/3*(b^2*c^5*d^2*x^6 - 3*b^2*c^3*d^2*x^4 + 3*b^2*c* 
d^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1) 
, c*x)/(c^2*x^2 - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.75 \[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{6} \, b^{2} c^{4} d^{2} x^{6} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a b c^{4} d^{2} x^{6} \arccos \left (c x\right ) + \frac {1}{6} \, a^{2} c^{4} d^{2} x^{6} - \frac {1}{108} \, b^{2} c^{4} d^{2} x^{6} - \frac {1}{18} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{2} x^{5} \arccos \left (c x\right ) - \frac {1}{18} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{2} x^{5} - \frac {1}{2} \, b^{2} c^{2} d^{2} x^{4} \arccos \left (c x\right )^{2} - a b c^{2} d^{2} x^{4} \arccos \left (c x\right ) - \frac {1}{2} \, a^{2} c^{2} d^{2} x^{4} + \frac {13}{288} \, b^{2} c^{2} d^{2} x^{4} + \frac {13}{72} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{2} x^{3} \arccos \left (c x\right ) + \frac {13}{72} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{2} x^{3} + \frac {1}{2} \, b^{2} d^{2} x^{2} \arccos \left (c x\right )^{2} + a b d^{2} x^{2} \arccos \left (c x\right ) + \frac {1}{2} \, a^{2} d^{2} x^{2} - \frac {11}{96} \, b^{2} d^{2} x^{2} - \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arccos \left (c x\right )}{48 \, c} - \frac {11 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{48 \, c} - \frac {11 \, b^{2} d^{2} \arccos \left (c x\right )^{2}}{96 \, c^{2}} - \frac {11 \, a b d^{2} \arccos \left (c x\right )}{48 \, c^{2}} + \frac {299 \, b^{2} d^{2}}{6912 \, c^{2}} \] Input:

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

Output:

1/6*b^2*c^4*d^2*x^6*arccos(c*x)^2 + 1/3*a*b*c^4*d^2*x^6*arccos(c*x) + 1/6* 
a^2*c^4*d^2*x^6 - 1/108*b^2*c^4*d^2*x^6 - 1/18*sqrt(-c^2*x^2 + 1)*b^2*c^3* 
d^2*x^5*arccos(c*x) - 1/18*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^2*x^5 - 1/2*b^2*c^ 
2*d^2*x^4*arccos(c*x)^2 - a*b*c^2*d^2*x^4*arccos(c*x) - 1/2*a^2*c^2*d^2*x^ 
4 + 13/288*b^2*c^2*d^2*x^4 + 13/72*sqrt(-c^2*x^2 + 1)*b^2*c*d^2*x^3*arccos 
(c*x) + 13/72*sqrt(-c^2*x^2 + 1)*a*b*c*d^2*x^3 + 1/2*b^2*d^2*x^2*arccos(c* 
x)^2 + a*b*d^2*x^2*arccos(c*x) + 1/2*a^2*d^2*x^2 - 11/96*b^2*d^2*x^2 - 11/ 
48*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arccos(c*x)/c - 11/48*sqrt(-c^2*x^2 + 1)*a 
*b*d^2*x/c - 11/96*b^2*d^2*arccos(c*x)^2/c^2 - 11/48*a*b*d^2*arccos(c*x)/c 
^2 + 299/6912*b^2*d^2/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^{2} \left (72 \mathit {acos} \left (c x \right )^{2} b^{2} c^{2} x^{2}-36 \mathit {acos} \left (c x \right )^{2} b^{2}-72 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c x +48 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}-144 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+144 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}+33 \mathit {asin} \left (c x \right ) a b -8 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}+26 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-33 \sqrt {-c^{2} x^{2}+1}\, a b c x +144 \left (\int \mathit {acos} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}-288 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+24 a^{2} c^{6} x^{6}-72 a^{2} c^{4} x^{4}+72 a^{2} c^{2} x^{2}-36 b^{2} c^{2} x^{2}\right )}{144 c^{2}} \] Input:

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

Output:

(d**2*(72*acos(c*x)**2*b**2*c**2*x**2 - 36*acos(c*x)**2*b**2 - 72*sqrt( - 
c**2*x**2 + 1)*acos(c*x)*b**2*c*x + 48*acos(c*x)*a*b*c**6*x**6 - 144*acos( 
c*x)*a*b*c**4*x**4 + 144*acos(c*x)*a*b*c**2*x**2 + 33*asin(c*x)*a*b - 8*sq 
rt( - c**2*x**2 + 1)*a*b*c**5*x**5 + 26*sqrt( - c**2*x**2 + 1)*a*b*c**3*x* 
*3 - 33*sqrt( - c**2*x**2 + 1)*a*b*c*x + 144*int(acos(c*x)**2*x**5,x)*b**2 
*c**6 - 288*int(acos(c*x)**2*x**3,x)*b**2*c**4 + 24*a**2*c**6*x**6 - 72*a* 
*2*c**4*x**4 + 72*a**2*c**2*x**2 - 36*b**2*c**2*x**2))/(144*c**2)