\(\int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a^2 d+(b^2+2 a d) x-(2 b+d) x^2+x^3)} \, dx\) [2872]

Optimal result

Integrand size = 77, antiderivative size = 306 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2 d+\left (b^2+2 a d\right ) x-(2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

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

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 10.95 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2 d+\left (b^2+2 a d\right ) x-(2 b+d) x^2+x^3\right )} \, dx=-\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x (-a+x) (-b+x)^2}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}}\right )-2 \log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}\right )+\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2-a \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+\left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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