\(\int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} (a^2 d+(b-2 a d) x+(-1+d) x^2)} \, dx\) [2630]

Optimal result

Integrand size = 63, antiderivative size = 232 \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{-2 b x+2 x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (b x-x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (b^2 x^2-2 b x^3+x^4+\left (-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \]

[Out]

3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)/(-2*b*x+2*x^2+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)
))/d^(2/3)+ln(b*x-x^2+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^(2/3)-1/2*ln(b^2*x^2-2*b*x^3+x^4+(-b*d^(1/3)*x+d
^(1/3)*x^2)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)+d^(2/3)*(a*b*x+(-a-b)*x^2+x^3)^(4/3))/d^(2/3)

Rubi [F]

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

[In]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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