\(\int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} (-a b+(a+b) x+(-1+d) x^2)} \, dx\) [2283]

Optimal result

Integrand size = 49, antiderivative size = 174 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]

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

Rubi [F]

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

[In]

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

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

Mathematica [A] (verified)

Time = 15.66 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.78 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]

[In]

Integrate[(-2*a*b*x + (a + b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(-(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^(2/3))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 d^{\frac {2}{3}}}\)	\(120\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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