\(\int \frac {(-a+x) (-b+x) (-2 a b x+(a+b) x^2)}{(x (-a+x) (-b+x))^{2/3} (a^2 b^2 d-2 a b (a+b) d x+(a^2+4 a b+b^2) d x^2-2 (a+b) d x^3+(-1+d) x^4)} \, dx\) [2846]

Optimal result

Integrand size = 92, antiderivative size = 291 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]

[Out]

-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)

Rubi [F]

\[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx \]

[In]

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]

[Out]

(3*(a + b)*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^6*(-a + x^3)^(1/3)*(-b + x^3)^(1/3
))/(a^2*b^2*d - 2*a^2*b*(1 + b/a)*d*x^3 + a^2*(1 + (b*(4*a + b))/a^2)*d*x^6 - 2*a*(1 + b/a)*d*x^9 - (1 - d)*x^
12), x], x, x^(1/3)])/((a - x)*(b - x)*x)^(2/3) + (6*a*b*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][De
fer[Int][(x^3*(-a + x^3)^(1/3)*(-b + x^3)^(1/3))/(-(a^2*b^2*d) + 2*a^2*b*(1 + b/a)*d*x^3 - a^2*(1 + (b*(4*a +
b))/a^2)*d*x^6 + 2*a*(1 + b/a)*d*x^9 + (1 - d)*x^12), x], x, x^(1/3)])/((a - x)*(b - x)*x)^(2/3)

Rubi steps \begin{align*} \text {integral}& = \int \frac {x (-a+x) (-b+x) (-2 a b+(a+b) x)}{(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 (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} (-2 a b+(a+b) x)}{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} \, dx}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-2 a b+(a+b) x^3\right )}{a^2 b^2 d-2 a b (a+b) d x^3+\left (a^2+4 a b+b^2\right ) d x^6-2 (a+b) d x^9+(-1+d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \left (\frac {(a+b) x^6 \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6-2 a \left (1+\frac {b}{a}\right ) d x^9-(1-d) x^{12}}+\frac {2 a b x^3 \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^3-a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6+2 a \left (1+\frac {b}{a}\right ) d x^9+(1-d) x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (6 a b x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^3-a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6+2 a \left (1+\frac {b}{a}\right ) d x^9+(1-d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{(x (-a+x) (-b+x))^{2/3}}+\frac {\left (3 (a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^6 \sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) d x^6-2 a \left (1+\frac {b}{a}\right ) d x^9-(1-d) x^{12}} \, dx,x,\sqrt [3]{x}\right )}{(x (-a+x) (-b+x))^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.71 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.73 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}}\right )-2 \log \left (x-\sqrt [6]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )-2 \log \left (x+\sqrt [6]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+\sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}\right )+\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+\sqrt [3]{d} (x (-a+x) (-b+x))^{2/3}\right )}{4 d^{2/3}} \]

[In]

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]

[Out]

-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)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.52

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}} x +\left (b -x \right ) \left (a -x \right ) \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{3}}{x^{3}}\right )}{4 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\)	\(152\)

[In]

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)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

[In]

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, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (2 \, a b x - {\left (a + b\right )} x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + {\left (d - 1\right )} x^{4} + {\left (a^{2} + 4 \, a b + b^{2}\right )} d x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \]

[In]

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, algorithm="maxima")

[Out]

-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)

Giac [A] (verification not implemented)

none

Time = 0.73 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.09 \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=-\frac {{\left | d \right |} \log \left ({\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{2 \, \left (-d^{5}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{4}} - \frac {\sqrt {3} \left (-d^{5}\right )^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{6}} - 2 \, {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}}}{\left (-\frac {1}{d}\right )^{\frac {1}{6}}}\right )}{2 \, d^{4}} - \frac {\left (-d^{5}\right )^{\frac {2}{3}} \log \left (\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{4}} - \frac {\left (-d^{5}\right )^{\frac {2}{3}} \log \left (-\sqrt {3} {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {1}{3}} \left (-\frac {1}{d}\right )^{\frac {1}{6}} + {\left (\frac {a b}{x^{2}} - \frac {a}{x} - \frac {b}{x} + 1\right )}^{\frac {2}{3}} + \left (-\frac {1}{d}\right )^{\frac {1}{3}}\right )}{4 \, d^{4}} \]

[In]

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, algorithm="giac")

[Out]

-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)*arcta
n(-(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

Mupad [F(-1)]

Timed out. \[ \int \frac {(-a+x) (-b+x) \left (-2 a b x+(a+b) x^2\right )}{(x (-a+x) (-b+x))^{2/3} \left (a^2 b^2 d-2 a b (a+b) d x+\left (a^2+4 a b+b^2\right ) d x^2-2 (a+b) d x^3+(-1+d) x^4\right )} \, dx=\int \frac {\left (x^2\,\left (a+b\right )-2\,a\,b\,x\right )\,\left (a-x\right )\,\left (b-x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^4\,\left (d-1\right )+a^2\,b^2\,d+d\,x^2\,\left (a^2+4\,a\,b+b^2\right )-2\,d\,x^3\,\left (a+b\right )-2\,a\,b\,d\,x\,\left (a+b\right )\right )} \,d x \]

[In]

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)

[Out]

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)