Integrand size = 84, antiderivative size = 250 \[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{4/3}+\left (a b+(-a-b) x+x^2\right )^{2/3} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )\right )}{4 d^{2/3}} \]
1/2*3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b+(-a-b)*x+x^2)^(2/3)/(2*a^2-4*a*x+2 *x^2+d^(1/3)*(a*b+(-a-b)*x+x^2)^(2/3)))/d^(2/3)+1/2*ln(a^2-2*a*x+x^2-d^(1/ 3)*(a*b+(-a-b)*x+x^2)^(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+d^(2/3)*(a*b+(-a-b)*x+x^2)^(4/3)+(a*b+(-a-b)*x+x^2)^(2/3)*(a^2*d^(1/3) -2*a*d^(1/3)*x+d^(1/3)*x^2))/d^(2/3)
Time = 17.71 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.97 \[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=-\frac {\sqrt [3]{(a-x) (b-x)} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 (a-x)^{4/3}}{\sqrt [3]{d} (b-x)^{2/3}}}{\sqrt {3}}\right )-2 \log \left ((a-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x}\right )-2 \log \left ((a-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x}\right )+\log \left ((a-x)^{4/3}-\sqrt [6]{d} (a-x)^{2/3} \sqrt [3]{b-x}+\sqrt [3]{d} (b-x)^{2/3}\right )+\log \left ((a-x)^{4/3}+\sqrt [6]{d} (a-x)^{2/3} \sqrt [3]{b-x}+\sqrt [3]{d} (b-x)^{2/3}\right )\right )}{4 d^{2/3} \sqrt [3]{a-x} \sqrt [3]{b-x}} \]
Integrate[((-b + x)*(-(a*(a - 2*b)) - 2*b*x + x^2))/(((-a + x)*(-b + x))^( 2/3)*(a^4 - b^2*d - 2*(2*a^3 - b*d)*x + (6*a^2 - d)*x^2 - 4*a*x^3 + x^4)), x]
-1/4*(((a - x)*(b - x))^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a - x)^(4/3))/(d^ (1/3)*(b - x)^(2/3)))/Sqrt[3]] - 2*Log[(a - x)^(2/3) - d^(1/6)*(b - x)^(1/ 3)] - 2*Log[(a - x)^(2/3) + d^(1/6)*(b - x)^(1/3)] + Log[(a - x)^(4/3) - d ^(1/6)*(a - x)^(2/3)*(b - x)^(1/3) + d^(1/3)*(b - x)^(2/3)] + Log[(a - x)^ (4/3) + d^(1/6)*(a - x)^(2/3)*(b - x)^(1/3) + d^(1/3)*(b - x)^(2/3)]))/(d^ (2/3)*(a - x)^(1/3)*(b - x)^(1/3))
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.
| \(\displaystyle \int \frac {(x-b) \left (-a (a-2 b)-2 b x+x^2\right )}{((x-a) (x-b))^{2/3} \left (a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {(x-b) \left (-a (a-2 b)-2 b x+x^2\right )}{\left (x (-a-b)+a b+x^2\right )^{2/3} \left (a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt [3]{(a-x) (b-x)} (a-2 b+x)}{a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4}dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {(a-2 b+x) \sqrt [3]{x (-a-b)+a b+x^2}}{a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {(a-2 b+x) \sqrt [3]{-x (a+b)+a b+x^2}}{a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 b \left (1-\frac {a}{2 b}\right ) \sqrt [3]{-x (a+b)+a b+x^2}}{-a^4+2 x \left (2 a^3-b d\right )-x^2 \left (6 a^2-d\right )+4 a x^3+b^2 d-x^4}+\frac {x \sqrt [3]{-x (a+b)+a b+x^2}}{a^4-2 x \left (2 a^3-b d\right )+x^2 \left (6 a^2-d\right )-4 a x^3-b^2 d+x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {x \sqrt [3]{x^2-(a+b) x+a b}}{a^4-4 x^3 a+x^4+\left (6 a^2-d\right ) x^2-b^2 d-2 \left (2 a^3-b d\right ) x}dx-(a-2 b) \int \frac {\sqrt [3]{x^2-(a+b) x+a b}}{-a^4+4 x^3 a-x^4-\left (6 a^2-d\right ) x^2+b^2 d+2 \left (2 a^3-b d\right ) x}dx\) |
Int[((-b + x)*(-(a*(a - 2*b)) - 2*b*x + x^2))/(((-a + x)*(-b + x))^(2/3)*( a^4 - b^2*d - 2*(2*a^3 - b*d)*x + (6*a^2 - d)*x^2 - 4*a*x^3 + x^4)),x]
3.28.23.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
\[\int \frac {\left (-b +x \right ) \left (-a \left (a -2 b \right )-2 b x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{4}-b^{2} d -2 \left (2 a^{3}-b d \right ) x +\left (6 a^{2}-d \right ) x^{2}-4 a \,x^{3}+x^{4}\right )}d x\]
int((-b+x)*(-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(a^4-b^2*d-2*(2*a^ 3-b*d)*x+(6*a^2-d)*x^2-4*a*x^3+x^4),x)
int((-b+x)*(-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(a^4-b^2*d-2*(2*a^ 3-b*d)*x+(6*a^2-d)*x^2-4*a*x^3+x^4),x)
Timed out. \[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate((-b+x)*(-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(a^4-b^2*d-2 *(2*a^3-b*d)*x+(6*a^2-d)*x^2-4*a*x^3+x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\text {Timed out} \]
integrate((-b+x)*(-a*(a-2*b)-2*b*x+x**2)/((-a+x)*(-b+x))**(2/3)/(a**4-b**2 *d-2*(2*a**3-b*d)*x+(6*a**2-d)*x**2-4*a*x**3+x**4),x)
\[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\int { \frac {{\left ({\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}\right )} {\left (b - x\right )}}{{\left (a^{4} - 4 \, a x^{3} + x^{4} - b^{2} d + {\left (6 \, a^{2} - d\right )} x^{2} - 2 \, {\left (2 \, a^{3} - b d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-b+x)*(-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(a^4-b^2*d-2 *(2*a^3-b*d)*x+(6*a^2-d)*x^2-4*a*x^3+x^4),x, algorithm="maxima")
integrate(((a - 2*b)*a + 2*b*x - x^2)*(b - x)/((a^4 - 4*a*x^3 + x^4 - b^2* d + (6*a^2 - d)*x^2 - 2*(2*a^3 - b*d)*x)*((a - x)*(b - x))^(2/3)), x)
\[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\int { \frac {{\left ({\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}\right )} {\left (b - x\right )}}{{\left (a^{4} - 4 \, a x^{3} + x^{4} - b^{2} d + {\left (6 \, a^{2} - d\right )} x^{2} - 2 \, {\left (2 \, a^{3} - b d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-b+x)*(-a*(a-2*b)-2*b*x+x^2)/((-a+x)*(-b+x))^(2/3)/(a^4-b^2*d-2 *(2*a^3-b*d)*x+(6*a^2-d)*x^2-4*a*x^3+x^4),x, algorithm="giac")
integrate(((a - 2*b)*a + 2*b*x - x^2)*(b - x)/((a^4 - 4*a*x^3 + x^4 - b^2* d + (6*a^2 - d)*x^2 - 2*(2*a^3 - b*d)*x)*((a - x)*(b - x))^(2/3)), x)
Timed out. \[ \int \frac {(-b+x) \left (-a (a-2 b)-2 b x+x^2\right )}{((-a+x) (-b+x))^{2/3} \left (a^4-b^2 d-2 \left (2 a^3-b d\right ) x+\left (6 a^2-d\right ) x^2-4 a x^3+x^4\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (-x^2+2\,b\,x+a\,\left (a-2\,b\right )\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2\,\left (d-6\,a^2\right )-2\,x\,\left (b\,d-2\,a^3\right )+b^2\,d+4\,a\,x^3-a^4-x^4\right )} \,d x \]
int(-((b - x)*(2*b*x + a*(a - 2*b) - x^2))/(((a - x)*(b - x))^(2/3)*(x^2*( d - 6*a^2) - 2*x*(b*d - 2*a^3) + b^2*d + 4*a*x^3 - a^4 - x^4)),x)