\(\int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} (a^2+(-2 a+b d) x+(1-d) x^2)} \, dx\) [2695]

Optimal result

Integrand size = 64, antiderivative size = 245 \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2+(-2 a+b d) x+(1-d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{-2 b \sqrt [3]{d} x+2 \sqrt [3]{d} x^2+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b \sqrt {d} x-\sqrt {d} x^2+\sqrt [6]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^2 d x^2-2 b d x^3+d x^4+\left (-b d^{2/3} x+d^{2/3} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

3^(1/2)*arctan(3^(1/2)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)/(-2*b*d^(1/3)*x+2*d^(1/3)*x^2+(a*b*x+(-a-b)*x^2+x^3)^(2/3)
))/d^(1/3)+ln(b*d^(1/2)*x-d^(1/2)*x^2+d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(2/3))/d^(1/3)-1/2*ln(b^2*d*x^2-2*b*d*x^3
+d*x^4+(-b*d^(2/3)*x+d^(2/3)*x^2)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(4/3))/d^(1/3)

Rubi [F]

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

[In]

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

Mathematica [F]

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

[In]

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

Maple [F]

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

[In]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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