\(\int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} (a^2 d+(b-2 a d) x+(-1+d) x^2)} \, dx\) [2636]

Optimal result

Integrand size = 51, antiderivative size = 233 \[ \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

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

[In]

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

[Out]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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