\(\int \frac {x (4 a b-3 (a+b) x+2 x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a b+(a+b) x-x^2+d x^4)} \, dx\) [2481]

Optimal result

Integrand size = 58, antiderivative size = 204 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 \sqrt [3]{d} x^2+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d} x^2+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^4+\sqrt [3]{d} x^2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 12.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a+x) (-b+x)}}{2 \sqrt [3]{d} x^2+\sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (-\sqrt [3]{d} x^2+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (d^{2/3} x^4+\sqrt [3]{d} x^2 \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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