Integrand size = 44, antiderivative size = 180 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*d^(1/3)*x/(d^(1/3)*x+2*(a*b*x^2+(-a-b)*x^3+x^4)^(1/ 3)))/d^(2/3)+ln(-d^(1/3)*x+(a*b*x^2+(-a-b)*x^3+x^4)^(1/3))/d^(2/3)-1/2*ln( d^(2/3)*x^2+d^(1/3)*x*(a*b*x^2+(-a-b)*x^3+x^4)^(1/3)+(a*b*x^2+(-a-b)*x^3+x ^4)^(2/3))/d^(2/3)
Time = 15.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.81 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \]
(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(x^2*(-a + x)*(-b + x ))^(1/3))] + 2*Log[-(d^(1/3)*x) + (x^2*(-a + x)*(-b + x))^(1/3)] - Log[d^( 2/3)*x^2 + d^(1/3)*x*(x^2*(-a + x)*(-b + x))^(1/3) + (x^2*(-a + x)*(-b + x ))^(2/3)])/(2*d^(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.
| \(\displaystyle \int \frac {x \left (x^2-a b\right )}{\left (x^2 (x-a) (x-b)\right )^{2/3} \left (-x (a+b+d)+a b+x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int -\frac {a b-x^2}{\sqrt [3]{x} \left (x^2-(a+b) x+a b\right )^{2/3} \left (x^2-(a+b+d) x+a b\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {a b-x^2}{\sqrt [3]{x} \left (x^2-(a+b) x+a b\right )^{2/3} \left (x^2-(a+b+d) x+a b\right )}dx}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {\sqrt [3]{x} \left (a b-x^2\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (x^2-(a+b+d) x+a b\right )}d\sqrt [3]{x}}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \left (\frac {\sqrt [3]{x} (2 a b-(a+b+d) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (x^2+(-a-b-d) x+a b\right )}-\frac {\sqrt [3]{x}}{\left (x^2-(a+b) x+a b\right )^{2/3}}\right )d\sqrt [3]{x}}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \left (\frac {(a+b+d) \left (\sqrt {a^2-2 a (b-d)+(b+d)^2}+a+b+d\right ) \int \frac {\sqrt [3]{x}}{\left (a+b+d-2 x+\sqrt {a^2-2 b a+2 d a+b^2+d^2+2 b d}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {4 a b \int \frac {\sqrt [3]{x}}{\left (a+b+d-2 x+\sqrt {a^2-2 b a+2 d a+b^2+d^2+2 b d}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2-2 a (b-d)+(b+d)^2}}+\frac {(a+b+d) \left (-\sqrt {a^2-2 a (b-d)+(b+d)^2}+a+b+d\right ) \int \frac {\sqrt [3]{x}}{\left (-a-b-d+2 x+\sqrt {a^2-2 b a+2 d a+b^2+d^2+2 b d}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {4 a b \int \frac {\sqrt [3]{x}}{\left (-a-b-d+2 x+\sqrt {a^2-2 b a+2 d a+b^2+d^2+2 b d}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {x^{2/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {2}{3},\frac {2}{3},\frac {5}{3},\frac {2 x}{a+b+\sqrt {a^2-2 b a+b^2}},\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}{2 \left (-x (a+b)+a b+x^2\right )^{2/3}}\right )}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\) |
3.24.19.3.1 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 = 1.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.72
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )}{2 d^{\frac {2}{3}}}\) | \(130\) |
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x+2*(x^2*(a-x)*(b-x))^(1/3))/d ^(1/3)/x)-ln((d^(2/3)*x^2+d^(1/3)*(x^2*(a-x)*(b-x))^(1/3)*x+(x^2*(a-x)*(b- x))^(2/3))/x^2)+2*ln((-d^(1/3)*x+(x^2*(a-x)*(b-x))^(1/3))/x))/d^(2/3)
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{3}} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )}} \,d x } \]
\[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{3}} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )}} \,d x } \]
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\int -\frac {x\,\left (a\,b-x^2\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2+\left (-a-b-d\right )\,x+a\,b\right )} \,d x \]