\(\int \frac {x^5 (a+b \arccos (c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [121]

Optimal result

Integrand size = 27, antiderivative size = 221 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {5 b x \sqrt {d-c^2 d x^2}}{3 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b x^3 \sqrt {d-c^2 d x^2}}{9 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^6 d^3}-\frac {b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{c^6 d^2 \sqrt {1-c^2 x^2}} \] Output:

-5/3*b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^2/(-c^2*x^2+1)^(1/2)-1/9*b*x^3*(-c^2*d 
*x^2+d)^(1/2)/c^3/d^2/(-c^2*x^2+1)^(1/2)+(a+b*arccos(c*x))/c^6/d/(-c^2*d*x 
^2+d)^(1/2)+2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c^6/d^2-1/3*(-c^2*d*x 
^2+d)^(3/2)*(a+b*arccos(c*x))/c^6/d^3-b*(-c^2*d*x^2+d)^(1/2)*arctanh(c*x)/ 
c^6/d^2/(-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 = 0.47 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (c \left (-b c x \sqrt {1-c^2 x^2} \left (15+c^2 x^2\right )+3 a \left (-8+4 c^2 x^2+c^4 x^4\right )+3 b \left (-8+4 c^2 x^2+c^4 x^4\right ) \arccos (c x)\right )-9 i b \sqrt {-c^2} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )\right )}{9 c^7 d^2 \left (-1+c^2 x^2\right )} \] Input:

Integrate[(x^5*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(3/2),x]
 

Output:

(Sqrt[d - c^2*d*x^2]*(c*(-(b*c*x*Sqrt[1 - c^2*x^2]*(15 + c^2*x^2)) + 3*a*( 
-8 + 4*c^2*x^2 + c^4*x^4) + 3*b*(-8 + 4*c^2*x^2 + c^4*x^4)*ArcCos[c*x]) - 
(9*I)*b*Sqrt[-c^2]*Sqrt[1 - c^2*x^2]*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] 
))/(9*c^7*d^2*(-1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5195, 27, 1467, 2009}

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.

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

\(\Big \downarrow \) 5195

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1467

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^5*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

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 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]
 

rule 2009

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

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 = 0.88 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.92

method	result	size
default	\(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )-\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{18 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x^{2}}{3 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {65 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arccos \left (c x \right )\right )}{72 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}\)	\(425\)
parts	\(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )-\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{18 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x^{2}}{3 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {65 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arccos \left (c x \right )\right )}{72 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}\)	\(425\)

Input:

int(x^5*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-x^2/c^2/d/(-c^2*d*x^2+d)^ 
(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2)))-31/18*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^ 
5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+5/3*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c 
^2*x^2-1)*arccos(c*x)*x^2-65/24*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2- 
1)*arccos(c*x)+1/24*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arccos(c* 
x)*cos(4*arccos(c*x))-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^6/ 
(c^2*x^2-1)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+b*(-d*(c^2*x^2-1))^(1/2)*(-c^2* 
x^2+1)^(1/2)/d^2/c^6/(c^2*x^2-1)*ln(I*(-c^2*x^2+1)^(1/2)+c*x-1)-1/72*b*(-d 
*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*sin(4*arccos(c*x))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.00 \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 12 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arccos \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{36 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}, \frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 6 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arccos \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{18 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}\right ] \] Input:

integrate(x^5*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas" 
)
 

Output:

[1/36*(9*(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*(b*c^3*x^3 + 15*b*c*x)*sqrt(- 
c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 12*(a*c^4*x^4 + 4*a*c^2*x^2 + (b*c^4*x 
^4 + 4*b*c^2*x^2 - 8*b)*arccos(c*x) - 8*a)*sqrt(-c^2*d*x^2 + d))/(c^8*d^2* 
x^2 - c^6*d^2), 1/18*(9*(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*(b*c^3*x^3 + 15* 
b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 6*(a*c^4*x^4 + 4*a*c^2*x^ 
2 + (b*c^4*x^4 + 4*b*c^2*x^2 - 8*b)*arccos(c*x) - 8*a)*sqrt(-c^2*d*x^2 + d 
))/(c^8*d^2*x^2 - c^6*d^2)]
 

Sympy [F]

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

integrate(x**5*(a+b*acos(c*x))/(-c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral(x**5*(a + b*acos(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^5*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima" 
)
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^(3/2),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. \[ \int \frac {x^5 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

int((x^5*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(3/2),x)
 

Output:

int((x^5*(a + b*acos(c*x)))/(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

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

int(x^5*(a+b*acos(c*x))/(-c^2*d*x^2+d)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

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