Integrand size = 76, antiderivative size = 343 \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{b}}}{(-1+x)^2}\right )}{2 b^{2/3}}-\frac {\log \left (\sqrt [6]{b}-\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{2/3}}-\frac {\log \left (-\sqrt [6]{b}+\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2+\left (\sqrt [6]{b}-\sqrt [6]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{4 b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-2 \sqrt [3]{b} x+\sqrt [3]{b} x^2+\left (-\sqrt [6]{b}+\sqrt [6]{b} x\right ) \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{4 b^{2/3}} \]
1/2*3^(1/2)*arctan((1/3*3^(1/2)-2/3*x*3^(1/2)+1/3*3^(1/2)*x^2+2/3*(x+(-1-k )*x^2+k*x^3)^(2/3)*3^(1/2)/b^(1/3))/(-1+x)^2)/b^(2/3)-1/2*ln(b^(1/6)-b^(1/ 6)*x+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(2/3)-1/2*ln(-b^(1/6)+b^(1/6)*x+(x+(-1- k)*x^2+k*x^3)^(1/3))/b^(2/3)+1/4*ln(b^(1/3)-2*b^(1/3)*x+b^(1/3)*x^2+(b^(1/ 6)-b^(1/6)*x)*(x+(-1-k)*x^2+k*x^3)^(1/3)+(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(2/ 3)+1/4*ln(b^(1/3)-2*b^(1/3)*x+b^(1/3)*x^2+(-b^(1/6)+b^(1/6)*x)*(x+(-1-k)*x ^2+k*x^3)^(1/3)+(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(2/3)
Time = 25.33 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.79 \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\frac {\sqrt [3]{k-\frac {1}{x}} (-1+x) \left (-2 \sqrt {3} \arctan \left (\frac {\left (k-\frac {1}{x}\right )^{2/3}+2 \sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}}{\sqrt {3} \left (k-\frac {1}{x}\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}-\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )-2 \log \left (\sqrt [3]{k-\frac {1}{x}}+\sqrt [6]{b} \left (-1+\frac {1}{x}\right )^{2/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}-\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )+\log \left (\left (k-\frac {1}{x}\right )^{2/3}+\sqrt [6]{b} \sqrt [3]{k-\frac {1}{x}} \left (-1+\frac {1}{x}\right )^{2/3}+\sqrt [3]{b} \left (-1+\frac {1}{x}\right )^{4/3}\right )\right )}{4 b^{2/3} \left (-1+\frac {1}{x}\right )^{2/3} \sqrt [3]{(-1+x) x (-1+k x)}} \]
Integrate[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/ 3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*b - 2*k)*x^3 + (-b + k^2)*x^4)),x]
((k - x^(-1))^(1/3)*(-1 + x)*(-2*Sqrt[3]*ArcTan[((k - x^(-1))^(2/3) + 2*b^ (1/3)*(-1 + x^(-1))^(4/3))/(Sqrt[3]*(k - x^(-1))^(2/3))] - 2*Log[(k - x^(- 1))^(1/3) - b^(1/6)*(-1 + x^(-1))^(2/3)] - 2*Log[(k - x^(-1))^(1/3) + b^(1 /6)*(-1 + x^(-1))^(2/3)] + Log[(k - x^(-1))^(2/3) - b^(1/6)*(k - x^(-1))^( 1/3)*(-1 + x^(-1))^(2/3) + b^(1/3)*(-1 + x^(-1))^(4/3)] + Log[(k - x^(-1)) ^(2/3) + b^(1/6)*(k - x^(-1))^(1/3)*(-1 + x^(-1))^(2/3) + b^(1/3)*(-1 + x^ (-1))^(4/3)]))/(4*b^(2/3)*(-1 + x^(-1))^(2/3)*((-1 + x)*x*(-1 + k*x))^(1/3 ))
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.
| \(\displaystyle \int \frac {\left (x^2-2 x+1\right ) ((2 k-1) x-1)}{\sqrt [3]{(1-x) x (1-k x)} \left (x^4 \left (k^2-b\right )+x^3 (4 b-2 k)+(1-6 b) x^2+4 b x-b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {((1-2 k) x+1) \left (x^2-2 x+1\right )}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (\left (b-k^2\right ) x^4-2 (2 b-k) x^3-(1-6 b) x^2-4 b x+b\right )}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {\sqrt [3]{x} ((1-2 k) x+1) \left (x^2-2 x+1\right )}{\sqrt [3]{k x^2-(k+1) x+1} \left (\left (b-k^2\right ) x^4-2 (2 b-k) x^3-(1-6 b) x^2-4 b x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(1-x)^2 \sqrt [3]{x} ((1-2 k) x+1)}{\sqrt [3]{k x^2-(k+1) x+1} \left (\left (b-k^2\right ) x^4-2 (2 b-k) x^3-(1-6 b) x^2-4 b x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \frac {(1-x)^{5/3} \sqrt [3]{x} ((1-2 k) x+1)}{\sqrt [3]{1-k x} \left (\left (b-k^2\right ) x^4-2 (2 b-k) x^3-(1-6 b) x^2-4 b x+b\right )}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \int \left (\frac {(1-2 k) x^{4/3} (1-x)^{5/3}}{\sqrt [3]{1-k x} \left (b \left (1-\frac {k^2}{b}\right ) x^4-4 b \left (1-\frac {k}{2 b}\right ) x^3-(1-6 b) x^2-4 b x+b\right )}+\frac {\sqrt [3]{x} (1-x)^{5/3}}{\sqrt [3]{1-k x} \left (b \left (1-\frac {k^2}{b}\right ) x^4-4 b \left (1-\frac {k}{2 b}\right ) x^3-(1-6 b) x^2-4 b x+b\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (\int \frac {(1-x)^{5/3} \sqrt [3]{x}}{\sqrt [3]{1-k x} \left (b (x-1)^4-x^2 (k x-1)^2\right )}d\sqrt [3]{x}+(1-2 k) \int \frac {(1-x)^{5/3} x^{4/3}}{\sqrt [3]{1-k x} \left (b (x-1)^4-x^2 (k x-1)^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\) |
Int[((-1 + (-1 + 2*k)*x)*(1 - 2*x + x^2))/(((1 - x)*x*(1 - k*x))^(1/3)*(-b + 4*b*x + (1 - 6*b)*x^2 + (4*b - 2*k)*x^3 + (-b + k^2)*x^4)),x]
3.30.34.3.1 Defintions of rubi rules used
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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])
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]
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]
\[\int \frac {\left (-1+\left (-1+2 k \right ) x \right ) \left (x^{2}-2 x +1\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +4 b x +\left (1-6 b \right ) x^{2}+\left (4 b -2 k \right ) x^{3}+\left (k^{2}-b \right ) x^{4}\right )}d x\]
int((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b) *x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x)
int((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+(1-6*b) *x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x)
Timed out. \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\text {Timed out} \]
integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+( 1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\text {Timed out} \]
integrate((-1+(-1+2*k)*x)*(x**2-2*x+1)/((1-x)*x*(-k*x+1))**(1/3)/(-b+4*b*x +(1-6*b)*x**2+(4*b-2*k)*x**3+(k**2-b)*x**4),x)
\[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (2 \, k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left ({\left (k^{2} - b\right )} x^{4} + 2 \, {\left (2 \, b - k\right )} x^{3} - {\left (6 \, b - 1\right )} x^{2} + 4 \, b x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+( 1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x, algorithm="maxima")
integrate(((2*k - 1)*x - 1)*(x^2 - 2*x + 1)/(((k^2 - b)*x^4 + 2*(2*b - k)* x^3 - (6*b - 1)*x^2 + 4*b*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
\[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\int { \frac {{\left ({\left (2 \, k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left ({\left (k^{2} - b\right )} x^{4} + 2 \, {\left (2 \, b - k\right )} x^{3} - {\left (6 \, b - 1\right )} x^{2} + 4 \, b x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-1+(-1+2*k)*x)*(x^2-2*x+1)/((1-x)*x*(-k*x+1))^(1/3)/(-b+4*b*x+( 1-6*b)*x^2+(4*b-2*k)*x^3+(k^2-b)*x^4),x, algorithm="giac")
integrate(((2*k - 1)*x - 1)*(x^2 - 2*x + 1)/(((k^2 - b)*x^4 + 2*(2*b - k)* x^3 - (6*b - 1)*x^2 + 4*b*x - b)*((k*x - 1)*(x - 1)*x)^(1/3)), x)
Timed out. \[ \int \frac {(-1+(-1+2 k) x) \left (1-2 x+x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+4 b x+(1-6 b) x^2+(4 b-2 k) x^3+\left (-b+k^2\right ) x^4\right )} \, dx=\int -\frac {\left (x\,\left (2\,k-1\right )-1\right )\,\left (x^2-2\,x+1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k^2\right )\,x^4+\left (2\,k-4\,b\right )\,x^3+\left (6\,b-1\right )\,x^2-4\,b\,x+b\right )} \,d x \]
int(-((x*(2*k - 1) - 1)*(x^2 - 2*x + 1))/((x*(k*x - 1)*(x - 1))^(1/3)*(b - x^3*(4*b - 2*k) + x^4*(b - k^2) - 4*b*x + x^2*(6*b - 1))),x)