Integrand size = 86, antiderivative size = 238 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 \left (2-2 x-2 k x+5 b x^2+2 k x^2\right ) \left (x-x^2-k x^2+k x^3\right )^{2/3}}{10 x^4}+\frac {\left (-\sqrt {3} a-\sqrt {3} b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {\left (-a-b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]
3/10*(5*b*x^2+2*k*x^2-2*k*x-2*x+2)*(k*x^3-k*x^2-x^2+x)^(2/3)/x^4+(-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^(1/3)+(b^2+a)*ln(-b^(1/3)*x+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(1/3) +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^(1/3)
Time = 10.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.81 \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\frac {3 ((-1+x) x (-1+k x))^{2/3} \left (2-2 (1+k) x+(5 b+2 k) x^2\right )}{10 x^4}-\frac {\sqrt {3} \left (a+b^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{(-1+x) x (-1+k x)}}\right )}{\sqrt [3]{b}}+\frac {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{(-1+x) x (-1+k x)}\right )}{\sqrt [3]{b}}-\frac {\left (a+b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 \sqrt [3]{b}} \]
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^4*((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k) *x + (-b + k)*x^2)),x]
(3*((-1 + x)*x*(-1 + k*x))^(2/3)*(2 - 2*(1 + k)*x + (5*b + 2*k)*x^2))/(10* x^4) - (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^(1/3) + ((a + b^2)*Log[-(b^(1/3)*x) + ((-1 + x)*x*(-1 + k*x))^(1/3)])/b^(1/3) - ((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^(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.
| \(\displaystyle \int \frac {((k+1) x-2) \left (x^4 \left (a+k^2\right )-2 \left (k^2+k\right ) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (x^2 (k-b)-(k+1) x+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int -\frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^{13/3} \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \sqrt [3]{k x^2-(k+1) x+1}}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^{13/3} \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \sqrt [3]{k x^2-(k+1) x+1}}dx}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \frac {(2-(k+1) x) \left (\left (k^2+a\right ) x^4-2 k (k+1) x^3+\left (k^2+4 k+1\right ) x^2-2 (k+1) x+1\right )}{x^{11/3} \left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \sqrt [3]{k x^2-(k+1) x+1}}d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (-\frac {(b+k) (k+1)}{x^{2/3} \sqrt [3]{k x^2-(k+1) x+1}}-\frac {3 (k+1)}{x^{8/3} \sqrt [3]{k x^2-(k+1) x+1}}+\frac {\left (b^2+a\right ) \sqrt [3]{x} (2-(k+1) x)}{\left (-\left ((b-k) x^2\right )-(k+1) x+1\right ) \sqrt [3]{k x^2-(k+1) x+1}}+\frac {k^2+4 k+2 b+1}{x^{5/3} \sqrt [3]{k x^2-(k+1) x+1}}+\frac {2}{x^{11/3} \sqrt [3]{k x^2-(k+1) x+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (\frac {(k+1) (b+k) \sqrt [3]{1-x} \sqrt [3]{1-k x} \sqrt [3]{1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {2}{3},-\frac {2 \left (\frac {k x}{k-\sqrt {(k-1)^2}+1}-\frac {k x}{k+\sqrt {(k-1)^2}+1}\right )}{1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}}\right )}{\sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1}}-\frac {\left (k^2+4 k+2 b+1\right ) \sqrt [3]{1-x} \sqrt [3]{1-k x} \left (1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}\right )^{2/3} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {5}{6}\right ) \left (\left (\frac {6 k x}{k-\sqrt {(k-1)^2}+1}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {2}{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 {4}{3},2,\frac {5}{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 )}{4 \sqrt [3]{2} \sqrt {\pi } x^{4/3} \sqrt [3]{1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}} \sqrt [3]{k x^2-(k+1) x+1} \operatorname {Gamma}\left (\frac {2}{3}\right )}+\frac {2 (k+1) \sqrt [3]{1-x} \sqrt [3]{1-k x} \left (1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}\right )^{2/3} \operatorname {Gamma}\left (-\frac {4}{3}\right ) \left (2 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {2}{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 {1}{3},1,\frac {2}{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 )^2-15 \sqrt {(k-1)^2} k x \operatorname {Hypergeometric2F1}\left (\frac {4}{3},2,\frac {5}{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 {4}{3},2,2;1,\frac {5}{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 {1}{3},1,\frac {2}{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 )-24 \sqrt {(k-1)^2} k^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},2,\frac {5}{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 {4}{3},2,2;1,\frac {5}{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 {4}{3},2,\frac {5}{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 {4}{3},2,2;1,\frac {5}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )\right )}{21 \left (k-\sqrt {(k-1)^2}+1\right )^3 x^{7/3} \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \sqrt [3]{1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}} \sqrt [3]{k x^2-(k+1) x+1} \operatorname {Gamma}\left (\frac {2}{3}\right )}+\frac {2 \sqrt [3]{1-x} \sqrt [3]{1-k x} \left (1-\frac {2 k x}{k-\sqrt {(k-1)^2}+1}\right )^{2/3} \operatorname {Gamma}\left (-\frac {7}{3}\right ) \left (14 \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {2}{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+36 k x \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {2}{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 )^3-117 \sqrt {(k-1)^2} k x \operatorname {Hypergeometric2F1}\left (\frac {4}{3},2,\frac {5}{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 \sqrt {(k-1)^2} k x \, _3F_2\left (\frac {4}{3},2,2;1,\frac {5}{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 {4}{3},2,2,2;1,1,\frac {5}{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 {1}{3},1,\frac {2}{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 )^2-198 \sqrt {(k-1)^2} k^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},2,\frac {5}{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 {4}{3},2,2;1,\frac {5}{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 {4}{3},2,2,2;1,1,\frac {5}{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 {1}{3},1,\frac {2}{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 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},2,\frac {5}{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 )-864 \sqrt {(k-1)^2} k^3 x^3 \, _3F_2\left (\frac {4}{3},2,2;1,\frac {5}{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 {4}{3},2,2,2;1,1,\frac {5}{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 {4}{3},2,\frac {5}{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 {4}{3},2,2;1,\frac {5}{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 {4}{3},2,2,2;1,1,\frac {5}{3};\frac {2 \sqrt {(k-1)^2} x}{2-\left (k-\sqrt {(k-1)^2}+1\right ) x}\right )\right )}{135 \left (k-\sqrt {(k-1)^2}+1\right )^4 x^{10/3} \left (-2 x k+k+\sqrt {(k-1)^2}+1\right ) \sqrt [3]{1-\frac {2 k x}{k+\sqrt {(k-1)^2}+1}} \sqrt [3]{k x^2-(k+1) x+1} \operatorname {Gamma}\left (\frac {2}{3}\right )}-\frac {4 \left (b^2+a\right ) (b-k) \int \frac {\sqrt [3]{x}}{\left (-k-2 (b-k) x-\sqrt {k^2-2 k+4 b+1}-1\right ) \sqrt [3]{k x^2+(-k-1) x+1}}d\sqrt [3]{x}}{\sqrt {(k-1)^2+4 b}}-\frac {4 \left (b^2+a\right ) (b-k) \int \frac {\sqrt [3]{x}}{\left (k+2 (b-k) x-\sqrt {k^2-2 k+4 b+1}+1\right ) \sqrt [3]{k x^2+(-k-1) x+1}}d\sqrt [3]{x}}{\sqrt {(k-1)^2+4 b}}-\frac {\left (b^2+a\right ) (k+1) \left (k-\sqrt {(k-1)^2+4 b}+1\right ) \int \frac {\sqrt [3]{x}}{\left (k+2 (b-k) x-\sqrt {k^2-2 k+4 b+1}+1\right ) \sqrt [3]{k x^2+(-k-1) x+1}}d\sqrt [3]{x}}{\sqrt {(k-1)^2+4 b}}+\frac {\left (b^2+a\right ) (k+1) \left (k+\sqrt {(k-1)^2+4 b}+1\right ) \int \frac {\sqrt [3]{x}}{\left (k+2 (b-k) x+\sqrt {k^2-2 k+4 b+1}+1\right ) \sqrt [3]{k x^2+(-k-1) x+1}}d\sqrt [3]{x}}{\sqrt {(k-1)^2+4 b}}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\) |
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^4*((1 - x)*x*(1 - k*x))^(1/3)*(1 - (1 + k)*x + ( -b + k)*x^2)),x]
3.27.62.3.1 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 = 1.18 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(-\frac {-3 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} b^{\frac {4}{3}} x^{2}-\frac {6 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}} \left (-1+x \right ) \left (k x -1\right ) b^{\frac {1}{3}}}{5}+x^{4} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (\left (-1+x \right ) x \left (k x -1\right )\right )^{\frac {1}{3}}}{x}\right )\right ) \left (b^{2}+a \right )}{2 b^{\frac {1}{3}} x^{4}}\) | \(170\) |
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 ^4/((1-x)*x*(-k*x+1))^(1/3)/(1-(1+k)*x+(-b+k)*x^2),x,method=_RETURNVERBOSE )
-1/2/b^(1/3)*(-3*((-1+x)*x*(k*x-1))^(2/3)*b^(4/3)*x^2-6/5*((-1+x)*x*(k*x-1 ))^(2/3)*(-1+x)*(k*x-1)*b^(1/3)+x^4*(-2*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^(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)-2*ln((-b^(1/3)*x+((-1+x )*x*(k*x-1))^(1/3))/x))*(b^2+a))/x^4
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
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^4/((1-x)*x*(-k*x+1))^(1/3)/(1-(1+k)*x+(-b+k)*x^2),x, algorithm="fri cas")
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\text {Timed out} \]
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**4/((1-x)*x*(-k*x+1))**(1/3)/(1-(1+k)*x+(-b+k)*x**2),x)
\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{4}} \,d x } \]
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^4/((1-x)*x*(-k*x+1))^(1/3)/(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)^(1/3)*((b - k)*x^2 + (k + 1)*x - 1)*x^4), x)
\[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int { -\frac {{\left ({\left (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 2 \, {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{4}} \,d x } \]
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^4/((1-x)*x*(-k*x+1))^(1/3)/(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)^(1/3)*((b - k)*x^2 + (k + 1)*x - 1)*x^4), x)
Timed out. \[ \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx=\int -\frac {\left (x\,\left (k+1\right )-2\right )\,\left (x^2\,\left (k^2+4\,k+1\right )-2\,x\,\left (k+1\right )+x^4\,\left (k^2+a\right )-2\,x^3\,\left (k^2+k\right )+1\right )}{x^4\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k\right )\,x^2+\left (k+1\right )\,x-1\right )} \,d x \]
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^4*(x*(k*x - 1)*(x - 1))^(1/3)*(x*(k + 1) + x^2*( b - k) - 1)),x)