\(\int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} (a b d-(a+b) d x+(-1+d) x^2)} \, dx\) [2225]

Optimal result

Integrand size = 51, antiderivative size = 166 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 15.58 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80 \[ \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}}\right )+2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )-\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+d^{2/3} (x (-a+x) (-b+x))^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )}{2 \left (\frac {1}{d}\right )^{\frac {2}{3}} d}\)	\(137\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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