\(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \arccos (c x)) \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 351 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2}}{768 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2}}{288 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{256 b c^3 \sqrt {1-c^2 x^2}} \] Output:

5/256*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-59/768*b*c*d^2*x 
^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+17/288*b*c^3*d^2*x^6*(-c^2*d*x^ 
2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/64*b*c^5*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^ 
2*x^2+1)^(1/2)-5/128*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^2+5/64 
*d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))+5/48*d*x^3*(-c^2*d*x^2+d)^ 
(3/2)*(a+b*arccos(c*x))+1/8*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))+5/2 
56*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.88 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {d^2 \left (-1440 b \sqrt {d-c^2 d x^2} \arccos (c x)^2-2880 a \sqrt {d} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-2880 a c x \sqrt {1-c^2 x^2}+22656 a c^3 x^3 \sqrt {1-c^2 x^2}-26112 a c^5 x^5 \sqrt {1-c^2 x^2}+9216 a c^7 x^7 \sqrt {1-c^2 x^2}+576 b \cos (2 \arccos (c x))+144 b \cos (4 \arccos (c x))-64 b \cos (6 \arccos (c x))+9 b \cos (8 \arccos (c x))\right )+24 b \sqrt {d-c^2 d x^2} \arccos (c x) (48 \sin (2 \arccos (c x))+24 \sin (4 \arccos (c x))-16 \sin (6 \arccos (c x))+3 \sin (8 \arccos (c x)))\right )}{73728 c^3 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
 

Output:

(d^2*(-1440*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 2880*a*Sqrt[d]*Sqrt[1 - 
c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt 
[d - c^2*d*x^2]*(-2880*a*c*x*Sqrt[1 - c^2*x^2] + 22656*a*c^3*x^3*Sqrt[1 - 
c^2*x^2] - 26112*a*c^5*x^5*Sqrt[1 - c^2*x^2] + 9216*a*c^7*x^7*Sqrt[1 - c^2 
*x^2] + 576*b*Cos[2*ArcCos[c*x]] + 144*b*Cos[4*ArcCos[c*x]] - 64*b*Cos[6*A 
rcCos[c*x]] + 9*b*Cos[8*ArcCos[c*x]]) + 24*b*Sqrt[d - c^2*d*x^2]*ArcCos[c* 
x]*(48*Sin[2*ArcCos[c*x]] + 24*Sin[4*ArcCos[c*x]] - 16*Sin[6*ArcCos[c*x]] 
+ 3*Sin[8*ArcCos[c*x]])))/(73728*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5203, 243, 49, 2009, 5203, 244, 2009, 5199, 15, 5211, 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.

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

\(\Big \downarrow \) 5203

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2dx}{8 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^2 \left (1-c^2 x^2\right )^2dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^6-2 c^2 x^4+x^2\right )dx^2}{16 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5203

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^3-c^2 x^5\right )dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5199

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int x^3dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))+\frac {\sqrt {d-c^2 d x^2} \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\right )+\frac {b c d \left (\frac {x^4}{4}-\frac {c^2 x^6}{6}\right ) \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}\right )+\frac {b c d^2 \left (\frac {c^4 x^8}{4}-\frac {2 c^2 x^6}{3}+\frac {x^4}{2}\right ) \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}\)

Input:

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]
 

Output:

(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^4/2 - (2*c^2*x^6)/3 + (c^4*x^8)/4))/(16*Sq 
rt[1 - c^2*x^2]) + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/8 + (5* 
d*((b*c*d*Sqrt[d - c^2*d*x^2]*(x^4/4 - (c^2*x^6)/6))/(6*Sqrt[1 - c^2*x^2]) 
 + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/6 + (d*((b*c*x^4*Sqrt[d 
 - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (x^3*Sqrt[d - c^2*d*x^2]*(a + b*Ar 
cCos[c*x]))/4 + (Sqrt[d - c^2*d*x^2]*(-1/4*(b*x^2)/c - (x*Sqrt[1 - c^2*x^2 
]*(a + b*ArcCos[c*x]))/(2*c^2) - (a + b*ArcCos[c*x])^2/(4*b*c^3)))/(4*Sqrt 
[1 - c^2*x^2])))/2))/8
 

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 49

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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 5203

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC 
os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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 = 0.83 (sec) , antiderivative size = 1080, normalized size of antiderivative = 3.08

method	result	size
default	\(\text {Expression too large to display}\)	\(1080\)
parts	\(\text {Expression too large to display}\)	\(1080\)

Input:

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
 

Output:

-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/19 
2*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/ 
128*a/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b 
*(5/256*(-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*d^2+1/16384*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x^7+128*I*(- 
c^2*x^2+1)^(1/2)*x^8*c^8+272*c^5*x^5-256*I*(-c^2*x^2+1)^(1/2)*x^6*c^6-88*c 
^3*x^3+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c*x-32*I*(-c^2*x^2+1)^(1/2)*x^2* 
c^2+I*(-c^2*x^2+1)^(1/2))*(I+8*arccos(c*x))*d^2/c^3/(c^2*x^2-1)-1/2304*(-d 
*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^6*c^6 
+38*c^3*x^3-48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-6*c*x+18*I*(-c^2*x^2+1)^(1/2)* 
x^2*c^2-I*(-c^2*x^2+1)^(1/2))*(I+6*arccos(c*x))*d^2/c^3/(c^2*x^2-1)+1/256* 
(-d*(c^2*x^2-1))^(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)*(-I+2*arccos(c*x))*d^2/c^3/(c^2*x^2-1)+1/16384*(-d*(c^ 
2*x^2-1))^(1/2)*(-128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+128*c^9*x^9+256*I*(-c^2 
*x^2+1)^(1/2)*x^6*c^6-320*c^7*x^7-160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+272*c^5 
*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-88*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+8*c*x 
)*(-I+8*arccos(c*x))*d^2/c^3/(c^2*x^2-1)-5/9216*(-d*(c^2*x^2-1))^(1/2)*(-I 
*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(I+12*arccos(c*x))*cos(5*arccos(c*x))*d 
^2/c^3/(c^2*x^2-1)-1/9216*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+ 
1)^(1/2)-I)*(13*I+12*arccos(c*x))*sin(5*arccos(c*x))*d^2/c^3/(c^2*x^2-1...
 

Fricas [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )} x^{2} \,d x } \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fricas" 
)
 

Output:

integral((a*c^4*d^2*x^6 - 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 - 2 
*b*c^2*d^2*x^4 + b*d^2*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\text {Timed out} \] Input:

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )} x^{2} \,d x } \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="maxima" 
)
 

Output:

b*sqrt(d)*integrate((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(c*x + 1)* 
sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x), x) + 1/384*(8*( 
-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d) + 10*(-c 
^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/ 
2)*arcsin(c*x)/c^3)*a
 

Giac [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.92 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {1}{8} \, \sqrt {-c^{2} d x^{2} + d} a c^{4} d^{2} x^{7} - \frac {17}{48} \, \sqrt {-c^{2} d x^{2} + d} a c^{2} d^{2} x^{5} + \frac {59}{192} \, \sqrt {-c^{2} d x^{2} + d} a d^{2} x^{3} - \frac {5 \, \sqrt {-c^{2} d x^{2} + d} a d^{2} x}{128 \, c^{2}} - \frac {5 \, a d^{3} \log \left ({\left | -c \sqrt {-d} x + \sqrt {c^{2} x^{2} - 1} \sqrt {-d} \right |}\right )}{128 \, c^{3} \sqrt {-d}} + \frac {1152 \, b c^{7} d^{\frac {5}{2}} x^{8} + 9216 \, \sqrt {-c^{2} x^{2} + 1} b c^{6} d^{\frac {5}{2}} x^{7} \arccos \left (c x\right ) - 4352 \, b c^{5} d^{\frac {5}{2}} x^{6} - 26112 \, \sqrt {-c^{2} x^{2} + 1} b c^{4} d^{\frac {5}{2}} x^{5} \arccos \left (c x\right ) + 5664 \, b c^{3} d^{\frac {5}{2}} x^{4} + 22656 \, \sqrt {-c^{2} x^{2} + 1} b c^{2} d^{\frac {5}{2}} x^{3} \arccos \left (c x\right ) - 1440 \, b c d^{\frac {5}{2}} x^{2} - 2880 \, \sqrt {-c^{2} x^{2} + 1} b d^{\frac {5}{2}} x \arccos \left (c x\right ) - \frac {1440 \, b d^{\frac {5}{2}} \arccos \left (c x\right )^{2}}{c} - \frac {359 \, b d^{\frac {5}{2}}}{c}}{73728 \, c^{2}} \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x)),x, algorithm="giac")
 

Output:

1/8*sqrt(-c^2*d*x^2 + d)*a*c^4*d^2*x^7 - 17/48*sqrt(-c^2*d*x^2 + d)*a*c^2* 
d^2*x^5 + 59/192*sqrt(-c^2*d*x^2 + d)*a*d^2*x^3 - 5/128*sqrt(-c^2*d*x^2 + 
d)*a*d^2*x/c^2 - 5/128*a*d^3*log(abs(-c*sqrt(-d)*x + sqrt(c^2*x^2 - 1)*sqr 
t(-d)))/(c^3*sqrt(-d)) + 1/73728*(1152*b*c^7*d^(5/2)*x^8 + 9216*sqrt(-c^2* 
x^2 + 1)*b*c^6*d^(5/2)*x^7*arccos(c*x) - 4352*b*c^5*d^(5/2)*x^6 - 26112*sq 
rt(-c^2*x^2 + 1)*b*c^4*d^(5/2)*x^5*arccos(c*x) + 5664*b*c^3*d^(5/2)*x^4 + 
22656*sqrt(-c^2*x^2 + 1)*b*c^2*d^(5/2)*x^3*arccos(c*x) - 1440*b*c*d^(5/2)* 
x^2 - 2880*sqrt(-c^2*x^2 + 1)*b*d^(5/2)*x*arccos(c*x) - 1440*b*d^(5/2)*arc 
cos(c*x)^2/c - 359*b*d^(5/2)/c)/c^2
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int(x^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int(x^2*(a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a +48 \sqrt {-c^{2} x^{2}+1}\, a \,c^{7} x^{7}-136 \sqrt {-c^{2} x^{2}+1}\, a \,c^{5} x^{5}+118 \sqrt {-c^{2} x^{2}+1}\, a \,c^{3} x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, a c x +384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{6}d x \right ) b \,c^{7}-768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) b \,c^{5}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) b \,c^{3}\right )}{384 c^{3}} \] Input:

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x)),x)
 

Output:

(sqrt(d)*d**2*(15*asin(c*x)*a + 48*sqrt( - c**2*x**2 + 1)*a*c**7*x**7 - 13 
6*sqrt( - c**2*x**2 + 1)*a*c**5*x**5 + 118*sqrt( - c**2*x**2 + 1)*a*c**3*x 
**3 - 15*sqrt( - c**2*x**2 + 1)*a*c*x + 384*int(sqrt( - c**2*x**2 + 1)*aco 
s(c*x)*x**6,x)*b*c**7 - 768*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*b 
*c**5 + 384*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*b*c**3))/(384*c** 
3)