\(\int \frac {-a b+(2 a-b) x}{\sqrt [3]{x (-a+x) (-b+x)} (-a^2+(2 a-b d) x+(-1+d) x^2)} \, dx\) [2566]

Optimal result

Integrand size = 56, antiderivative size = 218 \[ \int \frac {-a b+(2 a-b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2+(2 a-b d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}-\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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