\(\int \frac {x^5 (a+b \arccos (c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [132]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}+\frac {11 b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{6 c^6 d^3 \sqrt {1-c^2 x^2}} \] Output:

-1/6*b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^3/(-c^2*x^2+1)^(3/2)+b*x*(-c^2*d*x^2+d 
)^(1/2)/c^5/d^3/(-c^2*x^2+1)^(1/2)+1/3*(a+b*arccos(c*x))/c^6/d/(-c^2*d*x^2 
+d)^(3/2)-2*(a+b*arccos(c*x))/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-(-c^2*d*x^2+d)^ 
(1/2)*(a+b*arccos(c*x))/c^6/d^3+11/6*b*(-c^2*d*x^2+d)^(1/2)*arctanh(c*x)/c 
^6/d^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.78 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-\sqrt {-c^2} \left (b c x \left (5-6 c^2 x^2\right ) \sqrt {1-c^2 x^2}+2 a \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 b \left (8-12 c^2 x^2+3 c^4 x^4\right ) \arccos (c x)\right )+11 i b c \left (1-c^2 x^2\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )\right )}{6 c^6 \sqrt {-c^2} d^3 \left (-1+c^2 x^2\right )^2} \] Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(-(Sqrt[-c^2]*(b*c*x*(5 - 6*c^2*x^2)*Sqrt[1 - c^2*x^2 
] + 2*a*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*Ar 
cCos[c*x])) + (11*I)*b*c*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2 
]*x], 1]))/(6*c^6*Sqrt[-c^2]*d^3*(-1 + c^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5195, 27, 1471, 25, 299, 219}

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.

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

\(\Big \downarrow \) 5195

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int -\frac {3 c^4 x^4-12 c^2 x^2+8}{3 c^6 d^3 \left (1-c^2 x^2\right )^2}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \frac {3 c^4 x^4-12 c^2 x^2+8}{\left (1-c^2 x^2\right )^2}dx}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1471

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \left (-\frac {1}{2} \int -\frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \int \frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 299

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \left (11 \int \frac {1}{1-c^2 x^2}dx+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^3}-\frac {2 (a+b \arccos (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arccos (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \left (\frac {1}{2} \left (\frac {11 \text {arctanh}(c x)}{c}+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right ) \sqrt {d-c^2 d x^2}}{3 c^5 d^3 \sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(a + b*ArcCos[c*x])/(3*c^6*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcCos[c*x 
]))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x 
]))/(c^6*d^3) - (b*Sqrt[d - c^2*d*x^2]*(-1/2*x/(1 - c^2*x^2) + (6*x + (11* 
ArcTanh[c*x])/c)/2))/(3*c^5*d^3*Sqrt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 5195

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos 
[c*x])   u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.82 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.89

method	result	size
default	\(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-10 \arccos \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )\)	\(413\)
parts	\(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-10 \arccos \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{6 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )\)	\(413\)

Input:

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

Output:

a*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2 
/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^ 
2+1)^(1/2)*c*x+c^2*x^2-1)*(arccos(c*x)+I)/d^3/c^6/(c^2*x^2-1)-1/2*(-d*(c^2 
*x^2-1))^(1/2)*(-I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c*x)-I)/d^3/c 
^6/(c^2*x^2-1)+1/6*(-d*(c^2*x^2-1))^(1/2)*(12*c^2*x^2*arccos(c*x)+c*x*(-c^ 
2*x^2+1)^(1/2)-10*arccos(c*x))/c^6/(c^2*x^2-1)^2/d^3-11/6*(-d*(c^2*x^2-1)) 
^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*ln(I*(-c^2*x^2+1)^(1/2)+c*x- 
1)+11/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*ln(1 
+c*x+I*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.20 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [\frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arccos \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, -\frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arccos \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \] Input:

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

Output:

[1/24*(11*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(d)*log(-(c^6*d*x^6 + 5*c^4*d* 
x^4 - 5*c^2*d*x^2 + 4*(c^3*x^3 + c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 
 1)*sqrt(d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) + 4*(6*b*c^3*x^3 - 
 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 8*(3*a*c^4*x^4 - 12*a* 
c^2*x^2 + (3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*arccos(c*x) + 8*a)*sqrt(-c^2* 
d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3), -1/12*(11*(b*c^4*x^4 
 - 2*b*c^2*x^2 + b)*sqrt(-d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 
 1)*c*sqrt(-d)*x/(c^4*d*x^4 - d)) - 2*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d* 
x^2 + d)*sqrt(-c^2*x^2 + 1) + 4*(3*a*c^4*x^4 - 12*a*c^2*x^2 + (3*b*c^4*x^4 
 - 12*b*c^2*x^2 + 8*b)*arccos(c*x) + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3* 
x^4 - 2*c^8*d^3*x^2 + c^6*d^3)]
 

Sympy [F]

\[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral(x**5*(a + b*acos(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [F]

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

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

Output:

-1/3*a*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^(3 
/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d)) - 1/3*(3*(c^8*d^3*x^2 - c^6 
*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*integrate(1/3*(3*c^4*x^6 - 12*c 
^2*x^4 + 8*x^2)/(c^9*d^3*x^6 - 2*c^7*d^3*x^4 + c^5*d^3*x^2 + (c^7*d^3*x^4 
- 2*c^5*d^3*x^2 + c^3*d^3)*e^(log(c*x + 1) + log(-c*x + 1))), x) - (3*c^4* 
x^4 - 12*c^2*x^2 + 8)*sqrt(d)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))* 
b/((c^8*d^3*x^2 - c^6*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1))
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{8} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{6}+3 a \,c^{4} x^{4}-12 a \,c^{2} x^{2}+8 a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{6} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

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

Output:

(3*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**5)/(sqrt( - c**2*x**2 + 1)*c** 
4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b 
*c**8*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**5)/(sqrt( - c**2*x 
**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x** 
2 + 1)),x)*b*c**6 + 3*a*c**4*x**4 - 12*a*c**2*x**2 + 8*a)/(3*sqrt(d)*sqrt( 
 - c**2*x**2 + 1)*c**6*d**2*(c**2*x**2 - 1))