\(\int \frac {x (-a b+x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (a^2 b^2-2 a b (a+b) x+(a^2+4 a b+b^2-d) x^2-2 (a+b) x^3+x^4)} \, dx\) [2881]

Optimal result

Integrand size = 75, antiderivative size = 311 \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{4 d^{2/3}} \]

[Out]

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

\[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx \]

[In]

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

[Out]

(3*a*b*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^3/((-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(-
(a^2*b^2) + 2*a^2*b*(1 + b/a)*x^3 - a^2*(1 + (4*a*b + b^2 - d)/a^2)*x^6 + 2*a*(1 + b/a)*x^9 - x^12)), x], x, x
^(1/3)])/((a - x)*(b - x)*x^2)^(1/3) + (3*x^(2/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^9/((
-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(a^2*b^2 - 2*a^2*b*(1 + b/a)*x^3 + a^2*(1 + (4*a*b + b^2 - d)/a^2)*x^6 - 2*a*
(1 + b/a)*x^9 + x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {\sqrt [3]{x} \left (-a b+x^2\right )}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx}{\sqrt [3]{x^2 (-a+x) (-b+x)}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^3 \left (-a b+x^6\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a b (a+b) x^3+\left (a^2+4 a b+b^2-d\right ) x^6-2 (a+b) x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {a b x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-x^{12}\right )}+\frac {x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}}+\frac {\left (3 a b x^{2/3} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 b^2+2 a^2 b \left (1+\frac {b}{a}\right ) x^3-a^2 \left (1+\frac {4 a b+b^2-d}{a^2}\right ) x^6+2 a \left (1+\frac {b}{a}\right ) x^9-x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.69 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.71 \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}{\sqrt [3]{d} x^2}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [6]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-2 \log \left (\sqrt [6]{d} x+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )+\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )+\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{4 d^{2/3}} \]

[In]

Integrate[(x*(-(a*b) + x^2))/((x^2*(-a + x)*(-b + x))^(1/3)*(a^2*b^2 - 2*a*b*(a + b)*x + (a^2 + 4*a*b + b^2 -
d)*x^2 - 2*(a + b)*x^3 + x^4)),x]

[Out]

-1/4*(2*Sqrt[3]*ArcTan[(1 + (2*(x^2*(-a + x)*(-b + x))^(2/3))/(d^(1/3)*x^2))/Sqrt[3]] - 2*Log[-(d^(1/6)*x) + (
x^2*(-a + x)*(-b + x))^(1/3)] - 2*Log[d^(1/6)*x + (x^2*(-a + x)*(-b + x))^(1/3)] + Log[d^(1/3)*x^2 - d^(1/6)*x
*(x^2*(-a + x)*(-b + x))^(1/3) + (x^2*(-a + x)*(-b + x))^(2/3)] + Log[d^(1/3)*x^2 + d^(1/6)*x*(x^2*(-a + x)*(-
b + x))^(1/3) + (x^2*(-a + x)*(-b + x))^(2/3)])/d^(2/3)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{3}} \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}+\left (b -x \right ) \left (a -x \right ) \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{4 d^{\frac {2}{3}}}\)	\(144\)

[In]

int(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4),x,met
hod=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4)
,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(-a*b+x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(a**2*b**2-2*a*b*(a+b)*x+(a**2+4*a*b+b**2-d)*x**2-2*(a+b)*
x**3+x**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4)
,x, algorithm="maxima")

[Out]

-integrate((a*b - x^2)*x/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*((a
- x)*(b - x)*x^2)^(1/3)), x)

Giac [F]

\[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int { -\frac {{\left (a b - x^{2}\right )} x}{{\left (a^{2} b^{2} - 2 \, {\left (a + b\right )} a b x - 2 \, {\left (a + b\right )} x^{3} + x^{4} + {\left (a^{2} + 4 \, a b + b^{2} - d\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(-a*b+x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(a^2*b^2-2*a*b*(a+b)*x+(a^2+4*a*b+b^2-d)*x^2-2*(a+b)*x^3+x^4)
,x, algorithm="giac")

[Out]

integrate(-(a*b - x^2)*x/((a^2*b^2 - 2*(a + b)*a*b*x - 2*(a + b)*x^3 + x^4 + (a^2 + 4*a*b + b^2 - d)*x^2)*((a
- x)*(b - x)*x^2)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (-a b+x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a^2 b^2-2 a b (a+b) x+\left (a^2+4 a b+b^2-d\right ) x^2-2 (a+b) x^3+x^4\right )} \, dx=\int -\frac {x\,\left (a\,b-x^2\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^4-2\,x^3\,\left (a+b\right )+a^2\,b^2+x^2\,\left (a^2+4\,a\,b+b^2-d\right )-2\,a\,b\,x\,\left (a+b\right )\right )} \,d x \]

[In]

int(-(x*(a*b - x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4 - 2*x^3*(a + b) + a^2*b^2 + x^2*(4*a*b - d + a^2 + b^2)
- 2*a*b*x*(a + b))),x)

[Out]

int(-(x*(a*b - x^2))/((x^2*(a - x)*(b - x))^(1/3)*(x^4 - 2*x^3*(a + b) + a^2*b^2 + x^2*(4*a*b - d + a^2 + b^2)
- 2*a*b*x*(a + b))), x)