Integrand size = 91, antiderivative size = 387 \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [6]{d}-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (-a \sqrt [6]{d}+\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (-a \sqrt [6]{d}+\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]
-1/2*3^(1/2)*arctan((1/3*3^(1/2)*a^2-2/3*3^(1/2)*a*x+1/3*3^(1/2)*x^2+2/3*( a*b*x+(-a-b)*x^2+x^3)^(2/3)*3^(1/2)/d^(1/3))/(a-x)^2)/d^(2/3)+1/2*ln(a*d^( 1/6)-d^(1/6)*x+(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)+1/2*ln(-a*d^(1/6)+d^( 1/6)*x+(a*b*x+(-a-b)*x^2+x^3)^(1/3))/d^(2/3)-1/4*ln(a^2*d^(1/3)-2*a*d^(1/3 )*x+d^(1/3)*x^2+(a*d^(1/6)-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(a^2*d^(1/3)-2*a*d^(1/3)*x+d^(1/3)*x^ 2+(-a*d^(1/6)+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)
\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \]
Integrate[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*x + (-b^2 + 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)),x]
Integrate[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1/3)*(a^4*d - 4*a^3*d*x + (-b^2 + 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)), x]
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.
| \(\displaystyle \int \frac {\left (a^2-2 a x+x^2\right ) (x (2 a-b)-a b)}{\sqrt [3]{x (x-a) (x-b)} \left (a^4 d-4 a^3 d x+x^2 \left (6 a^2 d-b^2\right )+2 x^3 (b-2 a d)+(d-1) x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int -\frac {(a b-(2 a-b) x) \left (a^2-2 x a+x^2\right )}{\sqrt [3]{x} \sqrt [3]{x^2-(a+b) x+a b} \left (d a^4-4 d x a^3-(1-d) x^4+2 (b-2 a d) x^3-\left (b^2-6 a^2 d\right ) x^2\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(a b-(2 a-b) x) \left (a^2-2 x a+x^2\right )}{\sqrt [3]{x} \sqrt [3]{x^2-(a+b) x+a b} \left (d a^4-4 d x a^3-(1-d) x^4+2 (b-2 a d) x^3-\left (b^2-6 a^2 d\right ) x^2\right )}dx}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {\sqrt [3]{x} (a b-(2 a-b) x) \left (a^2-2 x a+x^2\right )}{\sqrt [3]{x^2-(a+b) x+a b} \left (d a^4-4 d x a^3-(1-d) x^4+2 (b-2 a d) x^3-\left (b^2-6 a^2 d\right ) x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{-x (a+b)+a b+x^2} \int \frac {(a-x)^2 \sqrt [3]{x} (a b-(2 a-b) x)}{\sqrt [3]{x^2-(a+b) x+a b} \left (d a^4-4 d x a^3-(1-d) x^4+2 (b-2 a d) x^3-\left (b^2-6 a^2 d\right ) x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \frac {(a-x)^{5/3} \sqrt [3]{x} (a b-(2 a-b) x)}{\sqrt [3]{b-x} \left (d a^4-4 d x a^3-(1-d) x^4+2 (b-2 a d) x^3-\left (b^2-6 a^2 d\right ) x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \int \left (\frac {(b-2 a) x^{4/3} (a-x)^{5/3}}{\sqrt [3]{b-x} \left (d a^4-4 d x a^3-(1-d) x^4+2 b \left (1-\frac {2 a d}{b}\right ) x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^2\right )}+\frac {a b \sqrt [3]{x} (a-x)^{5/3}}{\sqrt [3]{b-x} \left (d a^4-4 d x a^3-(1-d) x^4+2 b \left (1-\frac {2 a d}{b}\right ) x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x (a-x) (b-x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{a-x} \sqrt [3]{b-x} \left (a b \int \frac {(a-x)^{5/3} \sqrt [3]{x}}{\sqrt [3]{b-x} \left (d a^4-4 d x a^3-(1-d) x^4+2 b \left (1-\frac {2 a d}{b}\right ) x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^2\right )}d\sqrt [3]{x}-(2 a-b) \int \frac {(a-x)^{5/3} x^{4/3}}{\sqrt [3]{b-x} \left (d a^4-4 d x a^3-(1-d) x^4+2 b \left (1-\frac {2 a d}{b}\right ) x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x (a-x) (b-x)}}\) |
Int[((-(a*b) + (2*a - b)*x)*(a^2 - 2*a*x + x^2))/((x*(-a + x)*(-b + x))^(1 /3)*(a^4*d - 4*a^3*d*x + (-b^2 + 6*a^2*d)*x^2 + 2*(b - 2*a*d)*x^3 + (-1 + d)*x^4)),x]
3.30.84.3.1 Defintions of rubi rules used
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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]
\[\int \frac {\left (-a b +\left (2 a -b \right ) x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{4} d -4 a^{3} d x +\left (6 a^{2} d -b^{2}\right ) x^{2}+2 \left (-2 a d +b \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}d x\]
int((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3* d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x^3+(-1+d)*x^4),x)
int((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d-4*a^3* d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x^3+(-1+d)*x^4),x)
Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d- 4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x^3+(-1+d)*x^4),x, algorithm="fri cas")
Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate((-a*b+(2*a-b)*x)*(a**2-2*a*x+x**2)/(x*(-a+x)*(-b+x))**(1/3)/(a** 4*d-4*a**3*d*x+(6*a**2*d-b**2)*x**2+2*(-2*a*d+b)*x**3+(-1+d)*x**4),x)
\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d- 4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x^3+(-1+d)*x^4),x, algorithm="max ima")
-integrate((a^2 - 2*a*x + x^2)*(a*b - (2*a - b)*x)/((a^4*d - 4*a^3*d*x + ( d - 1)*x^4 - 2*(2*a*d - b)*x^3 + (6*a^2*d - b^2)*x^2)*((a - x)*(b - x)*x)^ (1/3)), x)
\[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a b - {\left (2 \, a - b\right )} x\right )}}{{\left (a^{4} d - 4 \, a^{3} d x + {\left (d - 1\right )} x^{4} - 2 \, {\left (2 \, a d - b\right )} x^{3} + {\left (6 \, a^{2} d - b^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]
integrate((-a*b+(2*a-b)*x)*(a^2-2*a*x+x^2)/(x*(-a+x)*(-b+x))^(1/3)/(a^4*d- 4*a^3*d*x+(6*a^2*d-b^2)*x^2+2*(-2*a*d+b)*x^3+(-1+d)*x^4),x, algorithm="gia c")
integrate(-(a^2 - 2*a*x + x^2)*(a*b - (2*a - b)*x)/((a^4*d - 4*a^3*d*x + ( d - 1)*x^4 - 2*(2*a*d - b)*x^3 + (6*a^2*d - b^2)*x^2)*((a - x)*(b - x)*x)^ (1/3)), x)
Timed out. \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=-\int \frac {\left (a\,b-x\,\left (2\,a-b\right )\right )\,\left (a^2-2\,a\,x+x^2\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (2\,x^3\,\left (b-2\,a\,d\right )+x^2\,\left (6\,a^2\,d-b^2\right )+a^4\,d+x^4\,\left (d-1\right )-4\,a^3\,d\,x\right )} \,d x \]
int(-((a*b - x*(2*a - b))*(a^2 - 2*a*x + x^2))/((x*(a - x)*(b - x))^(1/3)* (2*x^3*(b - 2*a*d) + x^2*(6*a^2*d - b^2) + a^4*d + x^4*(d - 1) - 4*a^3*d*x )),x)