Integrand size = 75, antiderivative size = 311 \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}} \]
1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^2/(d^(1/3)*x^2+2*(a*b*x^2+(-a-b)*x^3+ x^4)^(2/3)))/d^(2/3)+1/2*ln(-d^(1/6)*x+(a*b*x^2+(-a-b)*x^3+x^4)^(1/3))/d^( 2/3)+1/2*ln(d^(1/6)*x+(a*b*x^2+(-a-b)*x^3+x^4)^(1/3))/d^(2/3)-1/4*ln(d^(1/ 3)*x^2-d^(1/6)*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)-1/4*ln(d^(1/3)*x^2+d^(1/6)*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 = 12.69 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}{\sqrt [3]{d} x^2}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [6]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-2 \log \left (\sqrt [6]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )+\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )+\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{4 d^{2/3}} \]
Integrate[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(a^2*b^2 - 2*a *b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]
-1/4*(2*Sqrt[3]*ArcTan[(1 + (2*(x^2*(-a + x)*(-b + x))^(2/3))/(d^(1/3)*x^2 ))/Sqrt[3]] - 2*Log[-(d^(1/6)*x) + (x^2*(-a + x)*(-b + x))^(1/3)] - 2*Log[ d^(1/6)*x + (x^2*(-a + x)*(-b + x))^(1/3)] + Log[d^(1/3)*x^2 - d^(1/6)*x*( x^2*(-a + x)*(-b + x))^(1/3) + (x^2*(-a + x)*(-b + x))^(2/3)] + Log[d^(1/3 )*x^2 + d^(1/6)*x*(x^2*(-a + x)*(-b + x))^(1/3) + (x^2*(-a + x)*(-b + x))^ (2/3)])/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.
| \(\displaystyle \int \frac {x \left (x^2-a b\right )}{\sqrt [3]{x^2 (x-a) (x-b)} \left (x^2 \left (a^2+4 a b+b^2-d\right )+a^2 b^2-2 x^3 (a+b)-2 a b x (a+b)+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int -\frac {\sqrt [3]{x} \left (a b-x^2\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {\sqrt [3]{x} \left (a b-x^2\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {x \left (a b-x^2\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2-d\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \int \left (\frac {x^3}{\sqrt [3]{x^2-(a+b) x+a b} \left (-x^4+2 a \left (\frac {b}{a}+1\right ) x^3-a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) x-a^2 b^2\right )}+\frac {a b x}{\sqrt [3]{x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{-x (a+b)+a b+x^2} \left (\int \frac {x^3}{\sqrt [3]{x^2-(a+b) x+a b} \left (-x^4+2 a \left (\frac {b}{a}+1\right ) x^3-a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2+2 a^2 b \left (\frac {b}{a}+1\right ) x-a^2 b^2\right )}d\sqrt [3]{x}+a b \int \frac {x}{\sqrt [3]{x^2-(a+b) x+a b} \left (x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b^2+4 a b-d}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (a-x) (b-x)}}\) |
Int[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 - d)*x^2 - 2*(a + b)*x^3 + x^4)),x]
3.29.81.3.1 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]
Time = 0.72 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{3}} \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}+\left (b -x \right ) \left (a -x \right ) \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{4 d^{\frac {2}{3}}}\) | \(144\) |
int(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a *b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x,method=_RETURNVERBOSE)
1/4*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x^2+2*(x^2*(a-x)*(b-x))^(2/3)) /d^(1/3)/x^2)+2*ln((-d^(1/3)*x^2+(x^2*(a-x)*(b-x))^(2/3))/x^2)-ln((d^(1/3) *(x^2*(a-x)*(b-x))^(2/3)+(b-x)*(a-x)*(x^2*(a-x)*(b-x))^(1/3)+d^(2/3)*x^2)/ x^2))/d^(2/3)
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a ^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate(x*(-a*b+x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(a**2*b**2-2*a*b*(a+b) *x+(a**2+4*a*b+b**2-d)*x**2-2*(a+b)*x**3+x**4),x)
\[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a ^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="maxima")
-integrate((a*b - x^2)*x/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*((a - x)*(b - x)*x^2)^(1/3)), x)
\[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a ^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x, algorithm="giac")
integrate(-(a*b - x^2)*x/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*((a - x)*(b - x)*x^2)^(1/3)), x)
Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int -\frac {x\,\left (a\,b-x^2\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^4-2\,x^3\,\left (a+b\right )+a^2\,b^2+x^2\,\left (a^2+4\,a\,b+b^2-d\right )-2\,a\,b\,x\,\left (a+b\right )\right )} \,d x \]
int(-(x*(a*b - x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4 - 2*x^3*(a + b) + a ^2*b^2 + x^2*(4*a*b - d + a^2 + b^2) - 2*a*b*x*(a + b))),x)