\(\int \frac {x (-a b+x^2)}{(x^2 (-a+x) (-b+x))^{2/3} (a b-(a+b+d) x+x^2)} \, dx\) [2319]

Optimal result

Integrand size = 44, antiderivative size = 180 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 15.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.81 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )}{2 d^{\frac {2}{3}}}\)	\(130\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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