Integrand size = 80, antiderivative size = 297 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*x^2/(d^(1/3)*x^2+2*(a*b*x+(-a-b)*x^2+x ^3)^(2/3)))/d^(2/3)-1/2*ln(-d^(1/6)*x+(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3 )-1/2*ln(d^(1/6)*x+(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)+1/4*ln(d^(1/3)*x^ 2-d^(1/6)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)+(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^( 2/3)+1/4*ln(d^(1/3)*x^2+d^(1/6)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)+(a*b*x+(-a- b)*x^2+x^3)^(2/3))/d^(2/3)
Time = 10.85 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 (x (-a+x) (-b+x))^{2/3}}{\sqrt [3]{d} x^2}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )-2 \log \left (\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )+\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{4 d^{2/3}} \]
Integrate[(x^2*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-(a^2*b ^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d) *x^4)),x]
(2*Sqrt[3]*ArcTan[(1 + (2*(x*(-a + x)*(-b + x))^(2/3))/(d^(1/3)*x^2))/Sqrt [3]] - 2*Log[-(d^(1/6)*x) + (x*(-a + x)*(-b + x))^(1/3)] - 2*Log[d^(1/6)*x + (x*(-a + x)*(-b + x))^(1/3)] + Log[d^(1/3)*x^2 - d^(1/6)*x*(x*(-a + x)* (-b + x))^(1/3) + (x*(-a + x)*(-b + x))^(2/3)] + Log[d^(1/3)*x^2 + d^(1/6) *x*(x*(-a + x)*(-b + x))^(1/3) + (x*(-a + x)*(-b + x))^(2/3)])/(4*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.
| \(\displaystyle \int \frac {x^2 (x (a+b)-2 a b)}{\sqrt [3]{x (x-a) (x-b)} \left (-x^2 \left (a^2+4 a b+b^2\right )-a^2 b^2+2 x^3 (a+b)+2 a b x (a+b)+(d-1) x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {x^{5/3} (2 a b-(a+b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {x^{7/3} (2 a b-(a+b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 (a+b) x^3+\left (a^2+4 b a+b^2\right ) x^2-2 a b (a+b) x+a^2 b^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \left (\frac {(-a-b) x^{10/3}}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}+\frac {2 a b x^{7/3}}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \left (2 a b \int \frac {x^{7/3}}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}-(a+b) \int \frac {x^{10/3}}{\sqrt [3]{x^2-(a+b) x+a b} \left ((1-d) x^4-2 a \left (\frac {b}{a}+1\right ) x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) x^2-2 a^2 b \left (\frac {b}{a}+1\right ) x+a^2 b^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x (a-x) (b-x)}}\) |
Int[(x^2*(-2*a*b + (a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-(a^2*b^2) + 2*a*b*(a + b)*x - (a^2 + 4*a*b + b^2)*x^2 + 2*(a + b)*x^3 + (-1 + d)*x^4)) ,x]
3.29.54.3.1 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 = 0.54 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.45
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}} x +\left (b -x \right ) \left (a -x \right ) \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{3}}{x^{3}}\right )}{4 d^{\frac {2}{3}}}\) | \(135\) |
int(x^2*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^2*b^2+2*a*b*(a+b)*x-( a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x,method=_RETURNVERBOSE)
1/4*(2*3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x^2+2*(x*(a-x)*(b-x))^(2/3))/d^ (1/3)/x^2)-2*ln((-d^(1/3)*x^2+(x*(a-x)*(b-x))^(2/3))/x^2)+ln((d^(1/3)*(x*( a-x)*(b-x))^(2/3)*x+(b-x)*(a-x)*(x*(a-x)*(b-x))^(1/3)+d^(2/3)*x^3)/x^3))/d ^(2/3)
Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x^2*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x**2*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))**(1/3)/(-a**2*b**2+2*a*b *(a+b)*x-(a**2+4*a*b+b**2)*x**2+2*(a+b)*x**3+(-1+d)*x**4),x)
\[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{2}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate(x^2*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="maxima")
-integrate((2*a*b - (a + b)*x)*x^2/(((d - 1)*x^4 - a^2*b^2 + 2*(a + b)*a*b *x + 2*(a + b)*x^3 - (a^2 + 4*a*b + b^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
\[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b - {\left (a + b\right )} x\right )} x^{2}}{{\left ({\left (d - 1\right )} x^{4} - a^{2} b^{2} + 2 \, {\left (a + b\right )} a b x + 2 \, {\left (a + b\right )} x^{3} - {\left (a^{2} + 4 \, a b + b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate(x^2*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^2*b^2+2*a*b*(a+ b)*x-(a^2+4*a*b+b^2)*x^2+2*(a+b)*x^3+(-1+d)*x^4),x, algorithm="giac")
integrate(-(2*a*b - (a + b)*x)*x^2/(((d - 1)*x^4 - a^2*b^2 + 2*(a + b)*a*b *x + 2*(a + b)*x^3 - (a^2 + 4*a*b + b^2)*x^2)*((a - x)*(b - x)*x)^(1/3)), x)
Timed out. \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\int \frac {x^2\,\left (2\,a\,b-x\,\left (a+b\right )\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (2\,x^3\,\left (a+b\right )-x^2\,\left (a^2+4\,a\,b+b^2\right )-a^2\,b^2+x^4\,\left (d-1\right )+2\,a\,b\,x\,\left (a+b\right )\right )} \,d x \]
int(-(x^2*(2*a*b - x*(a + b)))/((x*(a - x)*(b - x))^(1/3)*(2*x^3*(a + b) - x^2*(4*a*b + a^2 + b^2) - a^2*b^2 + x^4*(d - 1) + 2*a*b*x*(a + b))),x)