\(\int \frac {1}{x \sqrt [3]{(-1+x) (q-2 q x+x^2)}} \, dx\) [43]

Optimal result

Integrand size = 21, antiderivative size = 118 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{q} (-1+x)}{\sqrt {3} \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}}\right )}{2 \sqrt [3]{q}}+\frac {\log (1-x)}{4 \sqrt [3]{q}}+\frac {\log (x)}{2 \sqrt [3]{q}}-\frac {3 \log \left (-\sqrt [3]{q} (-1+x)+\sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}\right )}{4 \sqrt [3]{q}} \] Output:

1/4*ln(1-x)/q^(1/3)+1/2*ln(x)/q^(1/3)-3/4*ln(-q^(1/3)*(-1+x)+((-1+x)*(-2*q 
*x+x^2+q))^(1/3))/q^(1/3)+1/2*arctan(1/3*3^(1/2)+2/3*q^(1/3)*(-1+x)/((-1+x 
)*(-2*q*x+x^2+q))^(1/3)*3^(1/2))*3^(1/2)/q^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.60 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\frac {\sqrt [3]{-1+x} \sqrt [3]{q-2 q x+x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{q} (-1+x)^{2/3}}{\sqrt [3]{q} (-1+x)^{2/3}+2 \sqrt [3]{q-2 q x+x^2}}\right )-2 \log \left (-\sqrt [3]{q} (-1+x)^{2/3}+\sqrt [3]{q-2 q x+x^2}\right )+\log \left (q^{2/3} (-1+x)^{4/3}+\sqrt [3]{q} (-1+x)^{2/3} \sqrt [3]{q-2 q x+x^2}+\left (q-2 q x+x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{q} \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \] Input:

Integrate[1/(x*((-1 + x)*(q - 2*q*x + x^2))^(1/3)),x]
 

Output:

((-1 + x)^(1/3)*(q - 2*q*x + x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*q^(1/3) 
*(-1 + x)^(2/3))/(q^(1/3)*(-1 + x)^(2/3) + 2*(q - 2*q*x + x^2)^(1/3))] - 2 
*Log[-(q^(1/3)*(-1 + x)^(2/3)) + (q - 2*q*x + x^2)^(1/3)] + Log[q^(2/3)*(- 
1 + x)^(4/3) + q^(1/3)*(-1 + x)^(2/3)*(q - 2*q*x + x^2)^(1/3) + (q - 2*q*x 
 + x^2)^(2/3)]))/(4*q^(1/3)*((-1 + x)*(q - 2*q*x + x^2))^(1/3))
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{x \sqrt [3]{(x-1) \left (-2 q x+q+x^2\right )}} \, dx\)

\(\Big \downarrow \) 2490

\(\displaystyle \int \frac {1}{\left (\frac {1}{3} (-2 q-1)+\frac {1}{3} (2 q+1)+x\right ) \sqrt [3]{\left (\frac {1}{3} (-2 q-1)+x\right )^3-\frac {1}{3} (1-4 q) (1-q) \left (\frac {1}{3} (-2 q-1)+x\right )-\frac {2}{27} (1-q)^2 (8 q+1)}}d\left (\frac {1}{3} (-2 q-1)+x\right )\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {1}{\left (\frac {1}{3} (-2 q-1)+\frac {1}{3} (2 q+1)+x\right ) \sqrt [3]{\left (\frac {1}{3} (-2 q-1)+x\right )^3-\frac {1}{3} (1-4 q) (1-q) \left (\frac {1}{3} (-2 q-1)+x\right )-\frac {2}{27} (1-q)^2 (8 q+1)}}d\left (\frac {1}{3} (-2 q-1)+x\right )\)

Input:

Int[1/(x*((-1 + x)*(q - 2*q*x + x^2))^(1/3)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \frac {1}{x {\left (\left (-1+x \right ) \left (-2 q x +x^{2}+q \right )\right )}^{\frac {1}{3}}}d x\]

Input:

int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)
 

Output:

int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (100) = 200\).

Time = 13.55 (sec) , antiderivative size = 1496, normalized size of antiderivative = 12.68 \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\text {Too large to display} \] Input:

integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="fricas")
 

Output:

[1/12*(sqrt(3)*q*sqrt((-q)^(1/3)/q)*log(-((q^3 - 30*q^2 - 51*q - 1)*x^6 + 
54*(q^3 + 6*q^2 + 2*q)*x^5 - 27*(17*q^3 + 26*q^2 + 2*q)*x^4 + 486*q^3*x + 
540*(2*q^3 + q^2)*x^3 - 81*q^3 - 135*(8*q^3 + q^2)*x^2 + 9*((2*q^2 - q - 1 
)*x^4 - 6*(q^2 - q)*x^3 + 3*(q^2 - q)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - 
 q)^(2/3)*(-q)^(1/3) + 9*((q^2 + 7*q + 1)*x^5 - (19*q^2 + 25*q + 1)*x^4 + 
9*(7*q^2 + 3*q)*x^3 + 45*q^2*x - 9*(9*q^2 + q)*x^2 - 9*q^2)*(-(2*q + 1)*x^ 
2 + x^3 + 3*q*x - q)^(1/3)*(-q)^(2/3) + sqrt(3)*(3*((4*q^2 + 13*q + 1)*x^4 
 - 6*(7*q^2 + 5*q)*x^3 - 72*q^2*x + 3*(31*q^2 + 5*q)*x^2 + 18*q^2)*(-(2*q 
+ 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^3 - 5*q^2 - 5*q)*x^5 
+ 5*(q^3 + 7*q^2 + q)*x^4 - 45*q^3*x - 45*(q^3 + q^2)*x^3 + 9*q^3 + 15*(5* 
q^3 + q^2)*x^2)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3) + ((q^3 + 24*q^2 
+ 3*q - 1)*x^6 - 54*(q^3 + 2*q^2)*x^5 + 81*(3*q^3 + 2*q^2)*x^4 - 162*q^3*x 
 - 108*(4*q^3 + q^2)*x^3 + 27*q^3 + 27*(14*q^3 + q^2)*x^2)*(-q)^(1/3))*sqr 
t((-q)^(1/3)/q))/x^6) - 2*(-q)^(2/3)*log(((-q)^(2/3)*(q - 1)*x^2 + 3*(-(2* 
q + 1)*x^2 + x^3 + 3*q*x - q)^(1/3)*(q*x - q)*(-q)^(1/3) + 3*(-(2*q + 1)*x 
^2 + x^3 + 3*q*x - q)^(2/3)*q)/x^2) + (-q)^(2/3)*log((3*((2*q + 1)*x^2 - 6 
*q*x + 3*q)*(-(2*q + 1)*x^2 + x^3 + 3*q*x - q)^(2/3)*(-q)^(2/3) + 3*((q^2 
+ 2*q)*x^3 + 9*q^2*x - (7*q^2 + 2*q)*x^2 - 3*q^2)*(-(2*q + 1)*x^2 + x^3 + 
3*q*x - q)^(1/3) - ((q^2 + 7*q + 1)*x^4 - 18*(q^2 + q)*x^3 - 36*q^2*x + 9* 
(5*q^2 + q)*x^2 + 9*q^2)*(-q)^(1/3))/x^4))/q, 1/12*(2*sqrt(3)*q*sqrt(-(...
 

Sympy [F]

\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int \frac {1}{x \sqrt [3]{\left (x - 1\right ) \left (- 2 q x + q + x^{2}\right )}}\, dx \] Input:

integrate(1/x/((-1+x)*(-2*q*x+x**2+q))**(1/3),x)
 

Output:

Integral(1/(x*((x - 1)*(-2*q*x + q + x**2))**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int { \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x} \,d x } \] Input:

integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="maxima")
 

Output:

integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x), x)
 

Giac [F]

\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int { \frac {1}{\left (-{\left (2 \, q x - x^{2} - q\right )} {\left (x - 1\right )}\right )^{\frac {1}{3}} x} \,d x } \] Input:

integrate(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((-(2*q*x - x^2 - q)*(x - 1))^(1/3)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int \frac {1}{x\,{\left (\left (x-1\right )\,\left (x^2-2\,q\,x+q\right )\right )}^{1/3}} \,d x \] Input:

int(1/(x*((x - 1)*(q - 2*q*x + x^2))^(1/3)),x)
 

Output:

int(1/(x*((x - 1)*(q - 2*q*x + x^2))^(1/3)), x)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx=\int \frac {1}{\left (-2 q \,x^{2}+x^{3}+3 q x -x^{2}-q \right )^{\frac {1}{3}} x}d x \] Input:

int(1/x/((-1+x)*(-2*q*x+x^2+q))^(1/3),x)
 

Output:

int(1/(( - 2*q*x**2 + 3*q*x - q + x**3 - x**2)**(1/3)*x),x)