Integrand size = 94, antiderivative size = 265 \[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{4 d^{2/3}} \]
1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)/(2*a^2-4*a *x+2*x^2+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d^(2/3)+1/2*ln(a^2-2*a*x+x ^2-d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^(2/3)-1/4*ln(a^4-4*a^3*x+6*a^2* x^2-4*a*x^3+x^4+(a^2*d^(1/3)-2*a*d^(1/3)*x+d^(1/3)*x^2)*(a*b*x+(-a-b)*x^2+ x^3)^(2/3)+d^(2/3)*(a*b*x+(-a-b)*x^2+x^3)^(4/3))/d^(2/3)
\[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \]
Integrate[(x*(-b + x)*(-(a^2*b) + 2*a^2*x + (-2*a + b)*x^2))/((x*(-a + x)* (-b + x))^(2/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]
Integrate[(x*(-b + x)*(-(a^2*b) + 2*a^2*x + (-2*a + b)*x^2))/((x*(-a + x)* (-b + x))^(2/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*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.
| \(\displaystyle \int \frac {x (x-b) \left (a^2 (-b)+2 a^2 x+x^2 (b-2 a)\right )}{(x (x-a) (x-b))^{2/3} \left (-a^4+4 a^3 x+x^2 \left (b^2 d-6 a^2\right )+2 x^3 (2 a-b d)+(d-1) x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int -\frac {(b-x) \sqrt [3]{x} \left (b a^2-2 x a^2+(2 a-b) x^2\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {(b-x) \sqrt [3]{x} \left (b a^2-2 x a^2+(2 a-b) x^2\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {(b-x) x \left (b a^2-2 x a^2+(2 a-b) x^2\right )}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 1395 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \int \frac {\sqrt [3]{b-x} x \left (b a^2-2 x a^2+(2 a-b) x^2\right )}{(a-x)^{2/3} \left (a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2\right )}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x (a b+(b-2 a) x)}{a^4-4 x a^3+(1-d) x^4-2 (2 a-b d) x^3+\left (6 a^2-b^2 d\right ) x^2}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \int \left (\frac {(b-2 a) \sqrt [3]{a-x} \sqrt [3]{b-x} x^2}{a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4}+\frac {a b \sqrt [3]{a-x} \sqrt [3]{b-x} x}{a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4}\right )d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} (a-x)^{2/3} (b-x)^{2/3} \left (a b \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x}{a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4}d\sqrt [3]{x}-(2 a-b) \int \frac {\sqrt [3]{a-x} \sqrt [3]{b-x} x^2}{a^4-4 x a^3+6 \left (1-\frac {b^2 d}{6 a^2}\right ) x^2 a^2-4 \left (1-\frac {b d}{2 a}\right ) x^3 a+(1-d) x^4}d\sqrt [3]{x}\right )}{(x (a-x) (b-x))^{2/3}}\) |
Int[(x*(-b + x)*(-(a^2*b) + 2*a^2*x + (-2*a + b)*x^2))/((x*(-a + x)*(-b + x))^(2/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]
3.28.73.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
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]
\[\int \frac {x \left (-b +x \right ) \left (-a^{2} b +2 a^{2} x +\left (-2 a +b \right ) x^{2}\right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (-a^{4}+4 a^{3} x +\left (b^{2} d -6 a^{2}\right ) x^{2}+2 \left (-b d +2 a \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}d x\]
int(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(-a^4+4 *a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x)
int(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(-a^4+4 *a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x)
Timed out. \[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/( -a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm=" fricas")
Timed out. \[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
integrate(x*(-b+x)*(-a**2*b+2*a**2*x+(-2*a+b)*x**2)/(x*(-a+x)*(-b+x))**(2/ 3)/(-a**4+4*a**3*x+(b**2*d-6*a**2)*x**2+2*(-b*d+2*a)*x**3+(-1+d)*x**4),x)
\[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { \frac {{\left (a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
integrate(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/( -a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm=" maxima")
integrate((a^2*b - 2*a^2*x + (2*a - b)*x^2)*(b - x)*x/(((d - 1)*x^4 - a^4 + 4*a^3*x - 2*(b*d - 2*a)*x^3 + (b^2*d - 6*a^2)*x^2)*((a - x)*(b - x)*x)^( 2/3)), x)
\[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { \frac {{\left (a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]
integrate(x*(-b+x)*(-a^2*b+2*a^2*x+(-2*a+b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/( -a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(-1+d)*x^4),x, algorithm=" giac")
integrate((a^2*b - 2*a^2*x + (2*a - b)*x^2)*(b - x)*x/(((d - 1)*x^4 - a^4 + 4*a^3*x - 2*(b*d - 2*a)*x^3 + (b^2*d - 6*a^2)*x^2)*((a - x)*(b - x)*x)^( 2/3)), x)
Timed out. \[ \int \frac {x (-b+x) \left (-a^2 b+2 a^2 x+(-2 a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=-\int -\frac {x\,\left (b-x\right )\,\left (x^2\,\left (2\,a-b\right )+a^2\,b-2\,a^2\,x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2\,\left (b^2\,d-6\,a^2\right )+2\,x^3\,\left (2\,a-b\,d\right )+4\,a^3\,x-a^4+x^4\,\left (d-1\right )\right )} \,d x \]
int((x*(b - x)*(x^2*(2*a - b) + a^2*b - 2*a^2*x))/((x*(a - x)*(b - x))^(2/ 3)*(x^2*(b^2*d - 6*a^2) + 2*x^3*(2*a - b*d) + 4*a^3*x - a^4 + x^4*(d - 1)) ),x)