3.29.20 \(\int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+(-1+b k^2) x^4)} \, dx\) [2820]

3.29.20.1 Optimal result

Integrand size = 68, antiderivative size = 279 \[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-2+2 x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-1+x}\right )}{b^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (-\sqrt [6]{b}+\sqrt [6]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}}{1-2 x+x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{5/6}} \]

output

1/2*3^(1/2)*arctan(3^(1/2)*b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(2-2*x+b^(1/ 
6)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^(5/6)-1/2*3^(1/2)*arctan(3^(1/2)*b^(1/6) 
*(x+(-1-k)*x^2+k*x^3)^(1/3)/(-2+2*x+b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b 
^(5/6)+arctanh(b^(1/6)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(-1+x))/b^(5/6)+1/2*arct 
anh((-b^(1/6)+b^(1/6)*x)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(1-2*x+x^2+b^(1/3)*(x+ 
(-1-k)*x^2+k*x^3)^(2/3)))/b^(5/6)
 

3.29.20.2 Mathematica [F]

\[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx \]

input

Integrate[(x*(-1 + k*x)*(-1 + (-1 + 2*k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)* 
(-1 + 4*x + (-6 + b)*x^2 + (4 - 2*b*k)*x^3 + (-1 + b*k^2)*x^4)),x]
 

output

Integrate[(x*(-1 + k*x)*(-1 + (-1 + 2*k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)* 
(-1 + 4*x + (-6 + b)*x^2 + (4 - 2*b*k)*x^3 + (-1 + b*k^2)*x^4)), x]
 

3.29.20.3 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.

\(\displaystyle \int \frac {x (k x-1) ((2 k-1) x-1)}{\sqrt [3]{(1-x) x (1-k x)} \left (x^4 \left (b k^2-1\right )+x^3 (4-2 b k)+(b-6) x^2+4 x-1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {x^{2/3} ((1-2 k) x+1) (1-k x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (\left (1-b k^2\right ) x^4-2 (2-b k) x^3+(6-b) x^2-4 x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {x^{2/3} ((1-2 k) x+1) (1-k x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (\left (1-b k^2\right ) x^4-2 (2-b k) x^3+(6-b) x^2-4 x+1\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {x^{4/3} ((1-2 k) x+1) (1-k x)}{\sqrt [3]{k x^2-(k+1) x+1} \left (\left (1-b k^2\right ) x^4-2 (2-b k) x^3+(6-b) x^2-4 x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 1395

\(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {x^{4/3} ((1-2 k) x+1) (1-k x)^{2/3}}{\sqrt [3]{1-x} \left (\left (1-b k^2\right ) x^4-2 (2-b k) x^3+(6-b) x^2-4 x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(1-2 k) (1-k x)^{2/3} x^{7/3}}{\sqrt [3]{1-x} \left (\left (1-b k^2\right ) x^4-4 \left (1-\frac {b k}{2}\right ) x^3+6 \left (1-\frac {b}{6}\right ) x^2-4 x+1\right )}+\frac {(1-k x)^{2/3} x^{4/3}}{\sqrt [3]{1-x} \left (\left (1-b k^2\right ) x^4-4 \left (1-\frac {b k}{2}\right ) x^3+6 \left (1-\frac {b}{6}\right ) x^2-4 x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (\int \frac {x^{4/3} (1-k x)^{2/3}}{\sqrt [3]{1-x} \left (\left (1-b k^2\right ) x^4-4 \left (1-\frac {b k}{2}\right ) x^3+6 \left (1-\frac {b}{6}\right ) x^2-4 x+1\right )}d\sqrt [3]{x}+(1-2 k) \int \frac {x^{7/3} (1-k x)^{2/3}}{\sqrt [3]{1-x} \left (\left (1-b k^2\right ) x^4-4 \left (1-\frac {b k}{2}\right ) x^3+6 \left (1-\frac {b}{6}\right ) x^2-4 x+1\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\)

input

Int[(x*(-1 + k*x)*(-1 + (-1 + 2*k)*x))/(((1 - x)*x*(1 - k*x))^(1/3)*(-1 + 
4*x + (-6 + b)*x^2 + (4 - 2*b*k)*x^3 + (-1 + b*k^2)*x^4)),x]
 

output

$Aborted
 

3.29.20.3.1 Defintions of rubi rules used

rule 25

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

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.29.20.4 Maple [F]

\[\int \frac {x \left (k x -1\right ) \left (-1+\left (-1+2 k \right ) x \right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-1+4 x +\left (-6+b \right ) x^{2}+\left (-2 b k +4\right ) x^{3}+\left (b \,k^{2}-1\right ) x^{4}\right )}d x\]

input

int(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(-1+4*x+(-6+b)*x^2+ 
(-2*b*k+4)*x^3+(b*k^2-1)*x^4),x)
 

output

int(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(-1+4*x+(-6+b)*x^2+ 
(-2*b*k+4)*x^3+(b*k^2-1)*x^4),x)
 

3.29.20.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(-1+4*x+(-6+b 
)*x^2+(-2*b*k+4)*x^3+(b*k^2-1)*x^4),x, algorithm="fricas")
 

output

Timed out
 

3.29.20.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\text {Timed out} \]

input

integrate(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))**(1/3)/(-1+4*x+(-6+ 
b)*x**2+(-2*b*k+4)*x**3+(b*k**2-1)*x**4),x)
 

output

Timed out
 

3.29.20.7 Maxima [F]

\[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (2 \, k - 1\right )} x - 1\right )} {\left (k x - 1\right )} x}{{\left ({\left (b k^{2} - 1\right )} x^{4} - 2 \, {\left (b k - 2\right )} x^{3} + {\left (b - 6\right )} x^{2} + 4 \, x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(-1+4*x+(-6+b 
)*x^2+(-2*b*k+4)*x^3+(b*k^2-1)*x^4),x, algorithm="maxima")
 

output

integrate(((2*k - 1)*x - 1)*(k*x - 1)*x/(((b*k^2 - 1)*x^4 - 2*(b*k - 2)*x^ 
3 + (b - 6)*x^2 + 4*x - 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
 

3.29.20.8 Giac [F]

\[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (2 \, k - 1\right )} x - 1\right )} {\left (k x - 1\right )} x}{{\left ({\left (b k^{2} - 1\right )} x^{4} - 2 \, {\left (b k - 2\right )} x^{3} + {\left (b - 6\right )} x^{2} + 4 \, x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(x*(k*x-1)*(-1+(-1+2*k)*x)/((1-x)*x*(-k*x+1))^(1/3)/(-1+4*x+(-6+b 
)*x^2+(-2*b*k+4)*x^3+(b*k^2-1)*x^4),x, algorithm="giac")
 

output

integrate(((2*k - 1)*x - 1)*(k*x - 1)*x/(((b*k^2 - 1)*x^4 - 2*(b*k - 2)*x^ 
3 + (b - 6)*x^2 + 4*x - 1)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
 

3.29.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+4 x+(-6+b) x^2+(4-2 b k) x^3+\left (-1+b k^2\right ) x^4\right )} \, dx=\int \frac {x\,\left (x\,\left (2\,k-1\right )-1\right )\,\left (k\,x-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b\,k^2-1\right )\,x^4+\left (4-2\,b\,k\right )\,x^3+\left (b-6\right )\,x^2+4\,x-1\right )} \,d x \]

input

int((x*(x*(2*k - 1) - 1)*(k*x - 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(4*x + x^ 
4*(b*k^2 - 1) - x^3*(2*b*k - 4) + x^2*(b - 6) - 1)),x)
 

output

int((x*(x*(2*k - 1) - 1)*(k*x - 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(4*x + x^ 
4*(b*k^2 - 1) - x^3*(2*b*k - 4) + x^2*(b - 6) - 1)), x)