Integrand size = 33, antiderivative size = 176 \[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-k x)}{\sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-(2-k) x)}{2^{2/3} \sqrt [3]{1-k}}+\frac {\log (1-k x)}{2\ 2^{2/3} \sqrt [3]{1-k}}-\frac {3 \log \left (-1+k x+2^{2/3} \sqrt [3]{1-k} \sqrt [3]{(1-x) x (1-k x)}\right )}{2\ 2^{2/3} \sqrt [3]{1-k}} \]
1/2*ln(1-(2-k)*x)*2^(1/3)/(1-k)^(1/3)+1/4*ln(-k*x+1)*2^(1/3)/(1-k)^(1/3)-3 /4*ln(-1+k*x+2^(2/3)*(1-k)^(1/3)*((1-x)*x*(-k*x+1))^(1/3))*2^(1/3)/(1-k)^( 1/3)-1/2*arctan(1/3*(1+2^(1/3)*(-k*x+1)/(1-k)^(1/3)/((1-x)*x*(-k*x+1))^(1/ 3))*3^(1/2))*3^(1/2)*2^(1/3)/(1-k)^(1/3)
\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=\int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx \]
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.
| \(\displaystyle \int \frac {1-k x}{((k-2) x+1) ((1-x) x (1-k x))^{2/3}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {1-k x}{x^{2/3} (1-(2-k) x) \left (k x^2-(k+1) x+1\right )^{2/3}}dx}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {1-k x}{(1-(2-k) x) \left (k x^2-(k+1) x+1\right )^{2/3}}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle \frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \int \frac {\sqrt [3]{1-k x}}{(1-x)^{2/3} (1-(2-k) x)}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 1028 |
| \(\displaystyle \frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \left (\frac {k \int \frac {1}{(1-x)^{2/3} (1-k x)^{2/3}}d\sqrt [3]{x}}{2-k}+\frac {2 (1-k) \int \frac {1}{(1-x)^{2/3} (1-(2-k) x) (1-k x)^{2/3}}d\sqrt [3]{x}}{2-k}\right )}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 905 |
| \(\displaystyle \frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \left (\frac {2 (1-k) \int \frac {1}{(1-x)^{2/3} (1-(2-k) x) (1-k x)^{2/3}}d\sqrt [3]{x}}{2-k}+\frac {k \sqrt [3]{x} \left (\frac {1-x}{1-k x}\right )^{2/3} \sqrt [3]{1-k x} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(1-k) x}{1-k x}\right )}{(2-k) (1-x)^{2/3}}\right )}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 1030 |
| \(\displaystyle \frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \left (\frac {2 (1-k) \int \left (\frac {1}{3 \left (1-\sqrt [3]{2-k} \sqrt [3]{x}\right ) (1-x)^{2/3} (1-k x)^{2/3}}+\frac {1}{3 \left (\sqrt [3]{-1} \sqrt [3]{2-k} \sqrt [3]{x}+1\right ) (1-x)^{2/3} (1-k x)^{2/3}}+\frac {1}{3 \left (1-(-1)^{2/3} \sqrt [3]{2-k} \sqrt [3]{x}\right ) (1-x)^{2/3} (1-k x)^{2/3}}\right )d\sqrt [3]{x}}{2-k}+\frac {k \sqrt [3]{x} \left (\frac {1-x}{1-k x}\right )^{2/3} \sqrt [3]{1-k x} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(1-k) x}{1-k x}\right )}{(2-k) (1-x)^{2/3}}\right )}{((1-x) x (1-k x))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \left (\frac {2 (1-k) \left (\frac {1}{3} \int \frac {1}{\left (1-\sqrt [3]{2-k} \sqrt [3]{x}\right ) (1-x)^{2/3} (1-k x)^{2/3}}d\sqrt [3]{x}+\frac {1}{3} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{2-k} \sqrt [3]{x}+1\right ) (1-x)^{2/3} (1-k x)^{2/3}}d\sqrt [3]{x}+\frac {1}{3} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{2-k} \sqrt [3]{x}\right ) (1-x)^{2/3} (1-k x)^{2/3}}d\sqrt [3]{x}\right )}{2-k}+\frac {k \sqrt [3]{x} \left (\frac {1-x}{1-k x}\right )^{2/3} \sqrt [3]{1-k x} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(1-k) x}{1-k x}\right )}{(2-k) (1-x)^{2/3}}\right )}{((1-x) x (1-k x))^{2/3}}\) |
3.1.45.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a + b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n) ^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(a*(c + d*x^n)))], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_ .)*(x_)^(n_))^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^n)^(p + 1)*(c + d* x^n)^(q - 1)*(e + f*x^n)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^n)^p *(c + d*x^n)^(q - 1)*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n, r }, x] && ILtQ[p, 0] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_ .)*(x_)^(n_))^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IGtQ[n, 0]
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]
\[\int \frac {-k x +1}{\left (1+\left (-2+k \right ) x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (132) = 264\).
Time = 54.28 (sec) , antiderivative size = 932, normalized size of antiderivative = 5.30 \[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=\text {Too large to display} \]
1/6*sqrt(3)*2^(1/3)*arctan(1/3*(24*sqrt(3)*2^(1/3)*((k^5 - 3*k^4 - 4*k^3 + 22*k^2 - 24*k + 8)*x^4 - 2*(k^4 - 10*k^3 + 27*k^2 - 26*k + 8)*x^3 - 6*(k^ 3 - 4*k^2 + 4*k - 1)*x^2 - 2*(k^2 - 1)*x + k - 1)*(k*x^3 - (k + 1)*x^2 + x )^(2/3)/(k - 1)^(1/3) - 6*sqrt(3)*2^(2/3)*((k^6 + 27*k^5 - 40*k^4 - 20*k^3 + 48*k^2 - 16*k)*x^5 - (33*k^5 + 55*k^4 - 220*k^3 + 132*k^2 + 16*k - 16)* x^4 + 2*(55*k^4 - 55*k^3 - 66*k^2 + 82*k - 16)*x^3 - 2*(55*k^3 - 99*k^2 + 38*k + 6)*x^2 + (33*k^2 - 61*k + 28)*x - k + 1)*(k*x^3 - (k + 1)*x^2 + x)^ (1/3)/(k - 1)^(2/3) + sqrt(3)*((k^6 - 48*k^5 - 192*k^4 + 416*k^3 - 48*k^2 - 192*k + 64)*x^6 + 6*(7*k^5 + 104*k^4 - 80*k^3 - 176*k^2 + 176*k - 32)*x^ 5 - 3*(139*k^4 + 256*k^3 - 768*k^2 + 352*k + 16)*x^4 + 4*(203*k^3 - 192*k^ 2 - 120*k + 104)*x^3 - 3*(139*k^2 - 208*k + 64)*x^2 + 6*(7*k - 8)*x + 1))/ ((k^6 + 96*k^5 - 48*k^4 - 160*k^3 + 240*k^2 - 192*k + 64)*x^6 - 6*(17*k^5 + 64*k^4 - 112*k^3 + 80*k^2 - 80*k + 32)*x^5 + 3*(149*k^4 + 32*k^3 - 96*k^ 2 - 160*k + 80)*x^4 - 4*(157*k^3 - 24*k^2 - 168*k + 40)*x^3 + 3*(149*k^2 - 128*k - 16)*x^2 - 6*(17*k - 16)*x + 1))/(k - 1)^(1/3) - 1/12*2^(1/3)*log( (12*2^(2/3)*(k*x^3 - (k + 1)*x^2 + x)^(2/3)*((k^3 + k^2 - 4*k + 2)*x^2 - 2 *(2*k^2 - 3*k + 1)*x + k - 1)/(k - 1)^(2/3) + 6*((k^3 + 8*k^2 - 8*k)*x^3 - (11*k^2 - 8)*x^2 + (11*k - 8)*x - 1)*(k*x^3 - (k + 1)*x^2 + x)^(1/3) + 2^ (1/3)*((k^4 + 28*k^3 - 12*k^2 - 32*k + 16)*x^4 - 4*(8*k^3 + 15*k^2 - 30*k + 8)*x^3 + 6*(13*k^2 - 10*k - 2)*x^2 - 4*(8*k - 7)*x + 1)/(k - 1)^(1/3)...
\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=- \int \frac {k x}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{k x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} - 2 x \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}} + \left (k x^{3} - k x^{2} - x^{2} + x\right )^{\frac {2}{3}}}\right )\, dx \]
-Integral(k*x/(k*x*(k*x**3 - k*x**2 - x**2 + x)**(2/3) - 2*x*(k*x**3 - k*x **2 - x**2 + x)**(2/3) + (k*x**3 - k*x**2 - x**2 + x)**(2/3)), x) - Integr al(-1/(k*x*(k*x**3 - k*x**2 - x**2 + x)**(2/3) - 2*x*(k*x**3 - k*x**2 - x* *2 + x)**(2/3) + (k*x**3 - k*x**2 - x**2 + x)**(2/3)), x)
\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=\int { -\frac {k x - 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (k - 2\right )} x + 1\right )}} \,d x } \]
\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=\int { -\frac {k x - 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (k - 2\right )} x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=-\int \frac {k\,x-1}{\left (x\,\left (k-2\right )+1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}} \,d x \]
\[ \int \frac {1-k x}{(1+(-2+k) x) ((1-x) x (1-k x))^{2/3}} \, dx=-\left (\int \frac {x}{x^{\frac {5}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}} k -2 x^{\frac {5}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}}+x^{\frac {2}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}}}d x \right ) k +\int \frac {1}{x^{\frac {5}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}} k -2 x^{\frac {5}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}}+x^{\frac {2}{3}} \left (k \,x^{2}-k x -x +1\right )^{\frac {2}{3}}}d x \]