\(\int \frac {(-b+x) (a b-2 b x+x^2)}{(x (-a+x) (-b+x)^2)^{2/3} (b d-(a+d) x+x^2)} \, dx\) [2840]

Optimal result

Integrand size = 50, antiderivative size = 290 \[ \int \frac {(-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} b \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{b \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (b \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-3*(b - x)*x*(1 - x/a)^(2/3)*(1 - x/b)^(1/3)*AppellF1[1/3, 2/3, 1/3, 4/3, x/a, x/b])/(-((a - x)*(b - x)^2*x))
^(2/3) + ((a - 2*b + d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Int][1/(
x^(2/3)*(-a + x)^(2/3)*(-b + x)^(1/3)*(-a - d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)*(b - x
)^2*x))^(2/3) + ((a - 2*b + d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[I
nt][1/(x^(2/3)*(-a + x)^(2/3)*(-b + x)^(1/3)*(-a - d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)
*(b - x)^2*x))^(2/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {a b-2 b x+x^2}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d-(a+d) x+x^2\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x}}+\frac {b (a-d)+(a-2 b+d) x}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d+(-a-d) x+x^2\right )}\right ) \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {b (a-d)+(a-2 b+d) x}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (b d+(-a-d) x+x^2\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )}+\frac {a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )}\right ) \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-b+x)^{4/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+x} \left (1-\frac {x}{a}\right )^{2/3}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (x^{2/3} (-b+x) \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}\right ) \int \frac {1}{x^{2/3} \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = -\frac {3 (b-x) x \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},\frac {1}{3},\frac {4}{3},\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{2/3}}+\frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{x^{2/3} (-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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