3.29.37 \(\int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} (1-4 k x+(-b+6 k^2) x^2+(2 b-4 k^3) x^3+(-b+k^4) x^4)} \, dx\) [2837]

3.29.37.1 Optimal result

Integrand size = 74, antiderivative size = 289 \[ \int \frac {(-1+x) x (1+(-2+k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (1-4 k x+\left (-b+6 k^2\right ) x^2+\left (2 b-4 k^3\right ) x^3+\left (-b+k^4\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-2+2 k x+\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{-1+k x}\right )}{b^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (-\sqrt [6]{b}+\sqrt [6]{b} k x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}}{1-2 k x+k^2 x^2+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{2 b^{5/6}} \]

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

3.29.37.2 Mathematica [F]

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

input

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

output

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

3.29.37.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.29.37.3.1 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] 
]
 

3.29.37.4 Maple [F]

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

input

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

output

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

3.29.37.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.29.37.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.29.37.7 Maxima [F]

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

input

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

output

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

3.29.37.8 Giac [F]

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

input

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

output

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

3.29.37.9 Mupad [F(-1)]

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

input

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

output

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