\(\int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} (b d-(a+d) x+x^2)} \, dx\) [2871]

Optimal result

Integrand size = 45, antiderivative size = 306 \[ \int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (b d-(a+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 b \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b \sqrt {d}-\sqrt {d} x+\sqrt [6]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^2 d-2 b d x+d x^2+\left (-b d^{2/3}+d^{2/3} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*x*(1 - x/a)^(1/3)*(1 - x/b)^(2/3)*AppellF1[2/3, 1/3, 2/3, 5/3, x/a, x/b])/(2*(-((a - x)*(b - x)^2*x))^(1/3)
) + ((a - 2*b + d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Int][1/(x^(1/
3)*(-a + x)^(1/3)*(-b + x)^(2/3)*(-a - d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)*(b - x)^2*x
))^(1/3) + ((a - 2*b + d - Sqrt[a^2 + 2*a*d - 4*b*d + d^2])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Int][1
/(x^(1/3)*(-a + x)^(1/3)*(-b + x)^(2/3)*(-a - d + Sqrt[a^2 + 2*a*d - 4*b*d + d^2] + 2*x)), x])/(-((a - x)*(b -
x)^2*x))^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {a b-2 b x+x^2}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b d-(a+d) x+x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}}+\frac {b (a-d)+(a-2 b+d) x}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b d+(-a-d) x+x^2\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {b (a-d)+(a-2 b+d) x}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (b d+(-a-d) x+x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \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}}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\sqrt [3]{x} (-b+x)^{2/3} \sqrt [3]{1-\frac {x}{a}}\right ) \int \frac {1}{\sqrt [3]{x} (-b+x)^{2/3} \sqrt [3]{1-\frac {x}{a}}} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3}} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ & = \frac {3 x \sqrt [3]{1-\frac {x}{a}} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {x}{a},\frac {x}{b}\right )}{2 \sqrt [3]{-\left ((a-x) (b-x)^2 x\right )}}+\frac {\left (\left (a-2 b+d-\sqrt {a^2+2 a d-4 b d+d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a-d+\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (\left (a-2 b+d+\sqrt {a^2+2 a d-4 b d+d^2}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a-d-\sqrt {a^2+2 a d-4 b d+d^2}+2 x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.55 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.71 \[ \int \frac {a b-2 b x+x^2}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (b d-(a+d) x+x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x (-a+x) (-b+x)^2}}{-2 b \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (-\sqrt [6]{d} \left (b \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}\right )\right )-\log \left (b^2 d-2 b d x+d x^2-b d^{2/3} \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} x \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

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

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(x*(-a + x)*(-b + x)^2)^(1/3))/(-2*b*d^(1/3) + 2*d^(1/3)*x + (x*(-a + x)*(-b + x)^2
)^(1/3))] + 2*Log[-(d^(1/6)*(b*d^(1/3) - d^(1/3)*x + (x*(-a + x)*(-b + x)^2)^(1/3)))] - Log[b^2*d - 2*b*d*x +
d*x^2 - b*d^(2/3)*(x*(-a + x)*(-b + x)^2)^(1/3) + d^(2/3)*x*(x*(-a + x)*(-b + x)^2)^(1/3) + d^(1/3)*(x*(-a + x
)*(-b + x)^2)^(2/3)])/(2*d^(1/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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