Integrand size = 92, antiderivative size = 291 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
-1/2*3^(1/2)*arctan(3^(1/2)*x^2/(x^2+2*d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3 )))/d^(2/3)+1/2*ln(x-d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)+1/2*ln( x+d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)-1/4*ln(x^2-d^(1/6)*x*(a*b* x+(-a-b)*x^2+x^3)^(1/3)+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^(2/3)-1/4* ln(x^2+d^(1/6)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)+d^(1/3)*(a*b*x+(-a-b)*x^2+x^ 3)^(2/3))/d^(2/3)
Time = 8.71 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.73 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}}\right )-2 \log \left (x-\sqrt [6]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )-2 \log \left (x+\sqrt [6]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+\sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}\right )+\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+\sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}\right )}{4 d^{2/3}} \]
Integrate[((-a + x)*(-b + x)*(-2*a*b*x + (a + b)*x^2))/((x*(-a + x)*(-b + x))^(2/3)*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (a^2 + 4*a*b + b^2)*d*x^2 - 2*( a + b)*d*x^3 + (-1 + d)*x^4)),x]
-1/4*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*d^(1/3)*(x*(-a + x)*(-b + x) )^(2/3))] - 2*Log[x - d^(1/6)*(x*(-a + x)*(-b + x))^(1/3)] - 2*Log[x + d^( 1/6)*(x*(-a + x)*(-b + x))^(1/3)] + Log[x^2 - d^(1/6)*x*(x*(-a + x)*(-b + x))^(1/3) + d^(1/3)*(x*(-a + x)*(-b + x))^(2/3)] + Log[x^2 + d^(1/6)*x*(x* (-a + x)*(-b + x))^(1/3) + d^(1/3)*(x*(-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.
| \(\displaystyle \int \frac {(x-a) (x-b) \left (x^2 (a+b)-2 a b x\right )}{(x (x-a) (x-b))^{2/3} \left (d x^2 \left (a^2+4 a b+b^2\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+(d-1) x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x (x-a) (x-b) (x (a+b)-2 a b)}{(x (x-a) (x-b))^{2/3} \left (d x^2 \left (a^2+4 a b+b^2\right )+a^2 b^2 d-2 d x^3 (a+b)-2 a b d x (a+b)+(d-1) x^4\right )}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int -\frac {(a-x) (b-x) \sqrt [3]{x} (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-\left ((1-d) x^4\right )-2 (a+b) d x^3+\left (a^2+4 b a+b^2\right ) d x^2-2 a b (a+b) d x+a^2 b^2 d\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {(a-x) (b-x) \sqrt [3]{x} (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-\left ((1-d) x^4\right )-2 (a+b) d x^3+\left (a^2+4 b a+b^2\right ) d x^2-2 a b (a+b) d x+a^2 b^2 d\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {(a-x) (b-x) x (2 a b-(a+b) x)}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (-\left ((1-d) x^4\right )-2 (a+b) d x^3+\left (a^2+4 b a+b^2\right ) d x^2-2 a b (a+b) d x+a^2 b^2 d\right )}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x (2 a b-(a+b) x)}{-\left ((1-d) x^4\right )-2 (a+b) d x^3+\left (a^2+4 b a+b^2\right ) d x^2-2 a b (a+b) d x+a^2 b^2 d}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \int \left (\frac {(-a-b) \sqrt [3]{a-x} \sqrt [3]{b-x} x^2}{-\left ((1-d) x^4\right )-2 a \left (\frac {b}{a}+1\right ) d x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) d x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d}+\frac {2 a b \sqrt [3]{a-x} \sqrt [3]{b-x} x}{-\left ((1-d) x^4\right )-2 a \left (\frac {b}{a}+1\right ) d x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) d x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d}\right )d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \left (2 a b \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x}{-\left ((1-d) x^4\right )-2 a \left (\frac {b}{a}+1\right ) d x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) d x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d}d\sqrt [3]{x}-(a+b) \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x^2}{-\left ((1-d) x^4\right )-2 a \left (\frac {b}{a}+1\right ) d x^3+a^2 \left (\frac {b (4 a+b)}{a^2}+1\right ) d x^2-2 a^2 b \left (\frac {b}{a}+1\right ) d x+a^2 b^2 d}d\sqrt [3]{x}\right )}{(x (a-x) (b-x))^{2/3}}\) |
Int[((-a + x)*(-b + x)*(-2*a*b*x + (a + b)*x^2))/((x*(-a + x)*(-b + x))^(2 /3)*(a^2*b^2*d - 2*a*b*(a + b)*d*x + (a^2 + 4*a*b + b^2)*d*x^2 - 2*(a + b) *d*x^3 + (-1 + d)*x^4)),x]
3.29.46.3.1 Defintions of rubi rules used
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_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = 152, normalized size of antiderivative = 0.52
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}} x +\left (b -x \right ) \left (a -x \right ) \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{3}}{x^{3}}\right )}{4 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\) | \(152\) |
int((-a+x)*(-b+x)*(-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*b^2*d- 2*a*b*(a+b)*d*x+(a^2+4*a*b+b^2)*d*x^2-2*(a+b)*d*x^3+(-1+d)*x^4),x,method=_ RETURNVERBOSE)
1/4*(2*3^(1/2)*arctan(1/3*3^(1/2)*((1/d)^(1/3)*x^2+2*(x*(a-x)*(b-x))^(2/3) )/(1/d)^(1/3)/x^2)+2*ln((-(1/d)^(1/3)*x^2+(x*(a-x)*(b-x))^(2/3))/x^2)-ln(( (1/d)^(1/3)*(x*(a-x)*(b-x))^(2/3)*x+(b-x)*(a-x)*(x*(a-x)*(b-x))^(1/3)+(1/d )^(2/3)*x^3)/x^3))/(1/d)^(1/3)/d
Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-b+x)*(-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2* b^2*d-2*a*b*(a+b)*d*x+(a^2+4*a*b+b^2)*d*x^2-2*(a+b)*d*x^3+(-1+d)*x^4),x, a lgorithm="fricas")
Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate((-a+x)*(-b+x)*(-2*a*b*x+(a+b)*x**2)/(x*(-a+x)*(-b+x))**(2/3)/(a* *2*b**2*d-2*a*b*(a+b)*d*x+(a**2+4*a*b+b**2)*d*x**2-2*(a+b)*d*x**3+(-1+d)*x **4),x)
\[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b x - {\left (a + b\right )} x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + {\left (d - 1\right )} x^{4} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
integrate((-a+x)*(-b+x)*(-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2* b^2*d-2*a*b*(a+b)*d*x+(a^2+4*a*b+b^2)*d*x^2-2*(a+b)*d*x^3+(-1+d)*x^4),x, a lgorithm="maxima")
-integrate((2*a*b*x - (a + b)*x^2)*(a - x)*(b - x)/((a^2*b^2*d - 2*(a + b) *a*b*d*x - 2*(a + b)*d*x^3 + (d - 1)*x^4 + (a^2 + 4*a*b + b^2)*d*x^2)*((a - x)*(b - x)*x)^(2/3)), x)
Time = 0.73 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.09 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {{\left | d \right |} \log \left ({\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{4}} - \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} - 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{4}} - \frac {\left (-d^{5}\right )^{\frac {2}{3}} \log \left (\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{4}} - \frac {\left (-d^{5}\right )^{\frac {2}{3}} \log \left (-\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{4}} \]
integrate((-a+x)*(-b+x)*(-2*a*b*x+(a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2* b^2*d-2*a*b*(a+b)*d*x+(a^2+4*a*b+b^2)*d*x^2-2*(a+b)*d*x^3+(-1+d)*x^4),x, a lgorithm="giac")
-1/2*abs(d)*log((a*b/x^2 - a/x - b/x + 1)^(2/3) + (-1/d)^(1/3))/(-d^5)^(1/ 3) + 1/2*sqrt(3)*(-d^5)^(2/3)*arctan((sqrt(3)*(-1/d)^(1/6) + 2*(a*b/x^2 - a/x - b/x + 1)^(1/3))/(-1/d)^(1/6))/d^4 - 1/2*sqrt(3)*(-d^5)^(2/3)*arctan( -(sqrt(3)*(-1/d)^(1/6) - 2*(a*b/x^2 - a/x - b/x + 1)^(1/3))/(-1/d)^(1/6))/ d^4 - 1/4*(-d^5)^(2/3)*log(sqrt(3)*(a*b/x^2 - a/x - b/x + 1)^(1/3)*(-1/d)^ (1/6) + (a*b/x^2 - a/x - b/x + 1)^(2/3) + (-1/d)^(1/3))/d^4 - 1/4*(-d^5)^( 2/3)*log(-sqrt(3)*(a*b/x^2 - a/x - b/x + 1)^(1/3)*(-1/d)^(1/6) + (a*b/x^2 - a/x - b/x + 1)^(2/3) + (-1/d)^(1/3))/d^4
Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\int \frac {\left (x^2\,\left (a+b\right )-2\,a\,b\,x\right )\,\left (a-x\right )\,\left (b-x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^4\,\left (d-1\right )+a^2\,b^2\,d+d\,x^2\,\left (a^2+4\,a\,b+b^2\right )-2\,d\,x^3\,\left (a+b\right )-2\,a\,b\,d\,x\,\left (a+b\right )\right )} \,d x \]
int(((x^2*(a + b) - 2*a*b*x)*(a - x)*(b - x))/((x*(a - x)*(b - x))^(2/3)*( x^4*(d - 1) + a^2*b^2*d + d*x^2*(4*a*b + a^2 + b^2) - 2*d*x^3*(a + b) - 2* a*b*d*x*(a + b))),x)