\(\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} (-a^4+4 a^3 x+(-6 a^2+b^2 d) x^2+2 (2 a-b d) x^3+(-1+d) x^4)} \, dx\) [2879]

Optimal result

Integrand size = 83, antiderivative size = 310 \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{2 a-2 x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a+2 x+\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{a-x}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}}{a^2-2 a x+x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{5/6}} \]

[Out]

-1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(2*a-2*x+d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)
))/d^(5/6)+1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(-2*a+2*x+d^(1/6)*(a*b*x+(-a-b)*x^2
+x^3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(a-x))/d^(5/6)+1/2*arctanh((a*d^(1/6)-d^(1/
6)*x)*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(a^2-2*a*x+x^2+d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d^(5/6)

Rubi [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \]

[In]

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

[Out]

(3*a*b*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x^4*(-b + x^3)^(2/3))/((-a + x^3)^(1/3)*
(-a^4 + 4*a^3*x^3 - 6*a^2*(1 - (b^2*d)/(6*a^2))*x^6 + 4*a*(1 - (b*d)/(2*a))*x^9 - (1 - d)*x^12)), x], x, x^(1/
3)])/((a - x)*(b - x)*x)^(1/3) + (3*(2*a - b)*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x
^7*(-b + x^3)^(2/3))/((-a + x^3)^(1/3)*(a^4 - 4*a^3*x^3 + 6*a^2*(1 - (b^2*d)/(6*a^2))*x^6 - 4*a*(1 - (b*d)/(2*
a))*x^9 + (1 - d)*x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x)^(1/3)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x^{2/3} (-b+x)^{2/3} (a b+(-2 a+b) x)}{\sqrt [3]{-a+x} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \left (-b+x^3\right )^{2/3} \left (a b+(-2 a+b) x^3\right )}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3+\left (-6 a^2+b^2 d\right ) x^6+2 (2 a-b d) x^9+(-1+d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \left (\frac {a b x^4 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 \left (1-\frac {b^2 d}{6 a^2}\right ) x^6+4 a \left (1-\frac {b d}{2 a}\right ) x^9-(1-d) x^{12}\right )}+\frac {(2 a-b) x^7 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 \left (1-\frac {b^2 d}{6 a^2}\right ) x^6-4 a \left (1-\frac {b d}{2 a}\right ) x^9+(1-d) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (3 (2 a-b) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^7 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (a^4-4 a^3 x^3+6 a^2 \left (1-\frac {b^2 d}{6 a^2}\right ) x^6-4 a \left (1-\frac {b d}{2 a}\right ) x^9+(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (3 a b \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^4 \left (-b+x^3\right )^{2/3}}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 \left (1-\frac {b^2 d}{6 a^2}\right ) x^6+4 a \left (1-\frac {b d}{2 a}\right ) x^9-(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx \]

[In]

Integrate[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^
2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)),x]

[Out]

Integrate[(x*(-b + x)*(a*b + (-2*a + b)*x))/((x*(-a + x)*(-b + x))^(1/3)*(-a^4 + 4*a^3*x + (-6*a^2 + b^2*d)*x^
2 + 2*(2*a - b*d)*x^3 + (-1 + d)*x^4)), x]

Maple [F]

\[\int \frac {x \left (-b +x \right ) \left (a b +\left (-2 a +b \right ) x \right )}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{4}+4 a^{3} x +\left (b^{2} d -6 a^{2}\right ) x^{2}+2 \left (-b d +2 a \right ) x^{3}+\left (-1+d \right ) x^{4}\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(
-1+d)*x^4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))**(1/3)/(-a**4+4*a**3*x+(b**2*d-6*a**2)*x**2+2*(-b*d+2*a)
*x**3+(-1+d)*x**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a b - {\left (2 \, a - b\right )} x\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(
-1+d)*x^4),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int { -\frac {{\left (a b - {\left (2 \, a - b\right )} x\right )} {\left (b - x\right )} x}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 2 \, {\left (b d - 2 \, a\right )} x^{3} + {\left (b^{2} d - 6 \, a^{2}\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(x*(-b+x)*(a*b+(-2*a+b)*x)/(x*(-a+x)*(-b+x))^(1/3)/(-a^4+4*a^3*x+(b^2*d-6*a^2)*x^2+2*(-b*d+2*a)*x^3+(
-1+d)*x^4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x (-b+x) (a b+(-2 a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^4+4 a^3 x+\left (-6 a^2+b^2 d\right ) x^2+2 (2 a-b d) x^3+(-1+d) x^4\right )} \, dx=\int -\frac {x\,\left (a\,b-x\,\left (2\,a-b\right )\right )\,\left (b-x\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2\,\left (b^2\,d-6\,a^2\right )+2\,x^3\,\left (2\,a-b\,d\right )+4\,a^3\,x-a^4+x^4\,\left (d-1\right )\right )} \,d x \]

[In]

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

[Out]

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