Integrand size = 86, antiderivative size = 235 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 \left (1-x-k x+4 b x^2+k x^2\right ) \sqrt [3]{x-x^2-k x^2+k x^3}}{4 x^3}+\frac {\left (\sqrt {3} a+\sqrt {3} b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}+\frac {\left (-a-b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 b^{2/3}} \]
3/4*(4*b*x^2+k*x^2-k*x-x+1)*(k*x^3-k*x^2-x^2+x)^(1/3)/x^3+(3^(1/2)*a+3^(1/ 2)*b^2)*arctan(3^(1/2)*b^(1/3)*x/(b^(1/3)*x+2*(x+(-1-k)*x^2+k*x^3)^(1/3))) /b^(2/3)+(b^2+a)*ln(-b^(1/3)*x+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(2/3)+1/2*(-b ^2-a)*ln(b^(2/3)*x^2+b^(1/3)*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 = 10.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.80 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 \sqrt [3]{(-1+x) x (-1+k x)} \left (1-(1+k) x+(4 b+k) x^2\right )}{4 x^3}+\frac {\sqrt {3} \left (a+b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{(-1+x) x (-1+k x)}}\right )}{b^{2/3}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{(-1+x) x (-1+k x)}\right )}{b^{2/3}}-\frac {\left (a+b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 b^{2/3}} \]
Integrate[((-2 + (1 + k)*x)*(1 - 2*(1 + k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2)*x^3 + (a + k^2)*x^4))/(x^3*((1 - x)*x*(1 - k*x))^(2/3)*(1 - (1 + k) *x + (-b + k)*x^2)),x]
(3*((-1 + x)*x*(-1 + k*x))^(1/3)*(1 - (1 + k)*x + (4*b + k)*x^2))/(4*x^3) + (Sqrt[3]*(a + b^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*((-1 + x)*x *(-1 + k*x))^(1/3))])/b^(2/3) + ((a + b^2)*Log[-(b^(1/3)*x) + ((-1 + x)*x* (-1 + k*x))^(1/3)])/b^(2/3) - ((a + b^2)*Log[b^(2/3)*x^2 + b^(1/3)*x*((-1 + x)*x*(-1 + k*x))^(1/3) + ((-1 + x)*x*(-1 + k*x))^(2/3)])/(2*b^(2/3))
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.
| \(\displaystyle \int \frac {((k+1) x-2) \left (x^4 \left (a+k^2\right )-2 \left (k^2+k\right ) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (x^2 (k-b)-(k+1) x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int -\frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^{11/3} \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \left (k x^2-(k+1) x+1\right )^{2/3}}dx}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^{11/3} \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \left (k x^2-(k+1) x+1\right )^{2/3}}dx}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^3 \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \left (k x^2-(k+1) x+1\right )^{2/3}}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \left (-\frac {\left (k^2+a\right ) (k+1)}{(k-b) \left (k x^2-(k+1) x+1\right )^{2/3}}-\frac {3 (k+1)}{x^2 \left (k x^2-(k+1) x+1\right )^{2/3}}+\frac {k^2+4 k+2 b+1}{x \left (k x^2-(k+1) x+1\right )^{2/3}}+\frac {\left (b^2+a\right ) \left (-k+\left (k^2+2 b+1\right ) x-1\right )}{(b-k) \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \left (k x^2-(k+1) x+1\right )^{2/3}}+\frac {2}{x^3 \left (k x^2-(k+1) x+1\right )^{2/3}}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \left (\frac {(k+1) \left (k^2+a\right ) (1-x)^{2/3} \sqrt [3]{x} (1-k x)^{2/3} \sqrt [3]{1-\frac {2 k x}{k-\sqrt {k^2-2 k+1}+1}} \left (\frac {1-\frac {2 k x}{k+\sqrt {k^2-2 k+1}+1}}{1-\frac {2 k x}{k-\sqrt {k^2-2 k+1}+1}}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {\sqrt {k^2-2 k+1} x}{1-\frac {2 k x}{k-\sqrt {k^2-2 k+1}+1}}\right )}{(b-k) \left (1-\frac {2 k x}{k+\sqrt {k^2-2 k+1}+1}\right )^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3}}-\frac {3 \left (k^2+4 k+2 b+1\right ) (1-x)^{2/3} (1-k x)^{2/3} \sqrt [3]{1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {7}{6}\right ) \left (\left (1-\frac {6 k x}{k-\sqrt {(k-1)^2}+1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )+\frac {12 \sqrt {(k-1)^2} k (1-x) x (1-k x) \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )}{\left (k-\sqrt {(k-1)^2}+1\right ) \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \left (2-\left (k-\sqrt {(k-1)^2}+1\right ) x\right )}\right )}{2^{2/3} \sqrt {\pi } x^{2/3} \left (1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}\right )^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \operatorname {Gamma}\left (\frac {1}{3}\right )}-\frac {2 (k+1) (1-x)^{2/3} (1-k x)^{2/3} \sqrt [3]{1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}} \operatorname {Gamma}\left (-\frac {2}{3}\right ) \left (\left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^3+6 k x \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2+3 \sqrt {(k-1)^2} k x \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2-9 \sqrt {(k-1)^2} k x \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2-36 k^2 x^2 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )+48 \sqrt {(k-1)^2} k^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )+36 \sqrt {(k-1)^2} k^2 x^2 \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )-108 \sqrt {(k-1)^2} k^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )-36 \sqrt {(k-1)^2} k^3 x^3 \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )\right )}{5 \left (k-\sqrt {(k-1)^2}+1\right )^3 x^{5/3} \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \left (1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}\right )^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \operatorname {Gamma}\left (\frac {1}{3}\right )}-\frac {(1-x)^{2/3} (1-k x)^{2/3} \sqrt [3]{1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}} \operatorname {Gamma}\left (-\frac {5}{3}\right ) \left (5 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^4+18 k x \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^3+9 \sqrt {(k-1)^2} k x \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^3-54 \sqrt {(k-1)^2} k x \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^3+27 \sqrt {(k-1)^2} k x \, _4F_3\left (\frac {5}{3},2,2,2;1,1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^3+108 k^2 x^2 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2+90 \sqrt {(k-1)^2} k^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2-108 \sqrt {(k-1)^2} k^2 x^2 \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2-162 \sqrt {(k-1)^2} k^2 x^2 \, _4F_3\left (\frac {5}{3},2,2,2;1,1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )^2-648 k^3 x^3 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )+972 \sqrt {(k-1)^2} k^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )+1080 \sqrt {(k-1)^2} k^3 x^3 \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )+324 \sqrt {(k-1)^2} k^3 x^3 \, _4F_3\left (\frac {5}{3},2,2,2;1,1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right ) \left (k-\sqrt {(k-1)^2}+1\right )-2376 \sqrt {(k-1)^2} k^4 x^4 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},2,\frac {7}{3},\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )-1296 \sqrt {(k-1)^2} k^4 x^4 \, _3F_2\left (\frac {5}{3},2,2;1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )-216 \sqrt {(k-1)^2} k^4 x^4 \, _4F_3\left (\frac {5}{3},2,2,2;1,1,\frac {7}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )\right )}{18 \left (k-\sqrt {(k-1)^2}+1\right )^4 x^{8/3} \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \left (1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}\right )^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \operatorname {Gamma}\left (\frac {1}{3}\right )}+\frac {\left (b^2+a\right ) \left (k^2+2 b+\sqrt {(k-1)^2+4 b} (k+1)+1\right ) \int \frac {1}{\left (-k+2 (k-b) x-\sqrt {k^2-2 k+4 b+1}-1\right ) \left (k x^2+(-k-1) x+1\right )^{2/3}}d\sqrt [3]{x}}{b-k}+\frac {\left (b^2+a\right ) \left (k^2+2 b-\sqrt {(k-1)^2+4 b} (k+1)+1\right ) \int \frac {1}{\left (-k+2 (k-b) x+\sqrt {k^2-2 k+4 b+1}-1\right ) \left (k x^2+(-k-1) x+1\right )^{2/3}}d\sqrt [3]{x}}{b-k}\right )}{((1-x) x (1-k x))^{2/3}}\) |
Int[((-2 + (1 + k)*x)*(1 - 2*(1 + k)*x + (1 + 4*k + k^2)*x^2 - 2*(k + k^2) *x^3 + (a + k^2)*x^4))/(x^3*((1 - x)*x*(1 - k*x))^(2/3)*(1 - (1 + k)*x + ( -b + k)*x^2)),x]
3.27.42.3.1 Defintions of rubi rules used
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[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(\frac {\left (\frac {3 \left (-1+x \right ) \left (k x -1\right ) b^{\frac {2}{3}}}{4}+3 b^{\frac {5}{3}} x^{2}\right ) \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}+x^{3} \left (-\arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (b^{2}+a \right )}{b^{\frac {2}{3}} x^{3}}\) | \(159\) |
int((-2+(1+k)*x)*(1-2*(1+k)*x+(k^2+4*k+1)*x^2-2*(k^2+k)*x^3+(k^2+a)*x^4)/x ^3/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x,method=_RETURNVERBOSE )
((3/4*(-1+x)*(k*x-1)*b^(2/3)+3*b^(5/3)*x^2)*((-1+x)*x*(k*x-1))^(1/3)+x^3*( -arctan(1/3*3^(1/2)*(b^(1/3)*x+2*((-1+x)*x*(k*x-1))^(1/3))/b^(1/3)/x)*3^(1 /2)+ln((-b^(1/3)*x+((-1+x)*x*(k*x-1))^(1/3))/x)-1/2*ln((b^(2/3)*x^2+b^(1/3 )*((-1+x)*x*(k*x-1))^(1/3)*x+((-1+x)*x*(k*x-1))^(2/3))/x^2))*(b^2+a))/b^(2 /3)/x^3
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(1-2*(1+k)*x+(k^2+4*k+1)*x^2-2*(k^2+k)*x^3+(k^2+a)* x^4)/x^3/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="fri cas")
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
integrate((-2+(1+k)*x)*(1-2*(1+k)*x+(k**2+4*k+1)*x**2-2*(k**2+k)*x**3+(k** 2+a)*x**4)/x**3/((1-x)*x*(-k*x+1))**(2/3)/(1-(1+k)*x+(-b+k)*x**2),x)
\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{3}} \,d x } \]
integrate((-2+(1+k)*x)*(1-2*(1+k)*x+(k^2+4*k+1)*x^2-2*(k^2+k)*x^3+(k^2+a)* x^4)/x^3/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="max ima")
-integrate(((k^2 + a)*x^4 - 2*(k^2 + k)*x^3 + (k^2 + 4*k + 1)*x^2 - 2*(k + 1)*x + 1)*((k + 1)*x - 2)/(((k*x - 1)*(x - 1)*x)^(2/3)*((b - k)*x^2 + (k + 1)*x - 1)*x^3), x)
\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{3}} \,d x } \]
integrate((-2+(1+k)*x)*(1-2*(1+k)*x+(k^2+4*k+1)*x^2-2*(k^2+k)*x^3+(k^2+a)* x^4)/x^3/((1-x)*x*(-k*x+1))^(2/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="gia c")
integrate(-((k^2 + a)*x^4 - 2*(k^2 + k)*x^3 + (k^2 + 4*k + 1)*x^2 - 2*(k + 1)*x + 1)*((k + 1)*x - 2)/(((k*x - 1)*(x - 1)*x)^(2/3)*((b - k)*x^2 + (k + 1)*x - 1)*x^3), x)
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^3 ((1-x) x (1-k x))^{2/3} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int -\frac {\left (x\,\left (k+1\right )-2\right )\,\left (x^2\,\left (k^2+4\,k+1\right )-2\,x\,\left (k+1\right )+x^4\,\left (k^2+a\right )-2\,x^3\,\left (k^2+k\right )+1\right )}{x^3\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b-k\right )\,x^2+\left (k+1\right )\,x-1\right )} \,d x \]
int(-((x*(k + 1) - 2)*(x^2*(4*k + k^2 + 1) - 2*x*(k + 1) + x^4*(a + k^2) - 2*x^3*(k + k^2) + 1))/(x^3*(x*(k*x - 1)*(x - 1))^(2/3)*(x*(k + 1) + x^2*( b - k) - 1)),x)