\(\int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} (-a b+(a+b) x+(-1+d) x^2)} \, dx\) [2271]

Optimal result

Integrand size = 46, antiderivative size = 173 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

3^(1/2)*arctan(3^(1/2)*d^(1/3)*x/(d^(1/3)*x+2*(a*b*x+(-a-b)*x^2+x^3)^(1/3)))/d^(1/3)-ln(-d^(1/3)*x+(a*b*x+(-a-
b)*x^2+x^3)^(1/3))/d^(1/3)+1/2*ln(d^(2/3)*x^2+d^(1/3)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)+(a*b*x+(-a-b)*x^2+x^3)^(2
/3))/d^(1/3)

Rubi [F]

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

[In]

Int[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]

[Out]

((a + b - Sqrt[a^2 - 2*a*b + b^2 + 4*a*b*d])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Int][1/(x^(1/3)*(-a +
x)^(1/3)*(-b + x)^(1/3)*(a + b - Sqrt[a^2 - 2*a*b + b^2 + 4*a*b*d] + 2*(-1 + d)*x)), x])/((a - x)*(b - x)*x)^
(1/3) + ((a + b + Sqrt[a^2 - 2*a*b + b^2 + 4*a*b*d])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Int][1/(x^(1/
3)*(-a + x)^(1/3)*(-b + x)^(1/3)*(a + b + Sqrt[a^2 - 2*a*b + b^2 + 4*a*b*d] + 2*(-1 + d)*x)), x])/((a - x)*(b
- x)*x)^(1/3)

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

Mathematica [A] (verified)

Time = 15.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

Integrate[(-2*a*b + (a + b)*x)/((x*(-a + x)*(-b + x))^(1/3)*(-(a*b) + (a + b)*x + (-1 + d)*x^2)),x]

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*x)/(d^(1/3)*x + 2*(x*(-a + x)*(-b + x))^(1/3))] - 2*Log[-(d^(1/3)*x) + (x*(
-a + x)*(-b + x))^(1/3)] + Log[d^(2/3)*x^2 + d^(1/3)*x*(x*(-a + x)*(-b + x))^(1/3) + (x*(-a + x)*(-b + x))^(2/
3)])/(2*d^(1/3))

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+\ln \left (\frac {-d^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}}{d^{\frac {1}{3}}}\)	\(119\)

[In]

int((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a*b+(a+b)*x+(-1+d)*x^2),x,method=_RETURNVERBOSE)

[Out]

-(3^(1/2)*arctan(1/3*3^(1/2)*(d^(1/3)*x+2*(x*(a-x)*(b-x))^(1/3))/d^(1/3)/x)+ln((-d^(1/3)*x+(x*(a-x)*(b-x))^(1/
3))/x)-1/2*ln((d^(2/3)*x^2+d^(1/3)*(x*(a-x)*(b-x))^(1/3)*x+(x*(a-x)*(b-x))^(2/3))/x^2))/d^(1/3)

Fricas [F(-1)]

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

[In]

integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a*b+(a+b)*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))**(1/3)/(-a*b+(a+b)*x+(-1+d)*x**2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a*b+(a+b)*x+(-1+d)*x^2),x, algorithm="maxima")

[Out]

-integrate((2*a*b - (a + b)*x)/(((a - x)*(b - x)*x)^(1/3)*((d - 1)*x^2 - a*b + (a + b)*x)), x)

Giac [F]

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

[In]

integrate((-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a*b+(a+b)*x+(-1+d)*x^2),x, algorithm="giac")

[Out]

integrate(-(2*a*b - (a + b)*x)/(((a - x)*(b - x)*x)^(1/3)*((d - 1)*x^2 - a*b + (a + b)*x)), x)

Mupad [F(-1)]

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

[In]

int(-(2*a*b - x*(a + b))/((x*(a - x)*(b - x))^(1/3)*(x*(a + b) - a*b + x^2*(d - 1))),x)

[Out]

-int((2*a*b - x*(a + b))/((x*(a - x)*(b - x))^(1/3)*(x*(a + b) - a*b + x^2*(d - 1))), x)