\(\int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a^2+(2 a+b^2 d) x-(1+2 b d) x^2+d x^3)} \, dx\) [2843]

Optimal result

Integrand size = 80, antiderivative size = 291 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[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 + (2*a + b^2*d)*x - (1 + 2
*b*d)*x^2 + d*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 - 2*a*(1 + (b^2*d)/(2*a))*x^3 + (1 + 2*b*d)*x^6 - d*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 + 2*a*(1 + (b^2*d)/(2*a))*x^3 - (1 + 2*b*d)*x^6 + d*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 + 2*a*(1 + (b^2*d)/(2*a))*x^3 - (1 + 2*b*d)*x^6 + d*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+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d 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+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d 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+\left (2 a+b^2 d\right ) x^3-(1+2 b d) x^6+d 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^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+2 b d) x^6-d x^9\right )}+\frac {a b x \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )}+\frac {x^7 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d 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+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d 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^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+2 b d) x^6-d 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+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.69 \[ \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx=-\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(b-x)^2 x (-a+x)}}{-2 a+2 x+\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}}\right )-2 \log \left (-a+x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}\right )+\log \left (a^2-2 a x+x^2-a \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)^2}+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+d^{2/3} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 d^{2/3}} \]

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

[Out]

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

Maple [F]

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

[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+(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*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+(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*x^3),x)

Fricas [F(-1)]

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

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

[Out]

Timed out

Sympy [F(-1)]

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

[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+(b**2*d+2*a)*x-(2*b*d+1)*x**2
+d*x**3),x)

[Out]

Timed out

Maxima [F]

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

[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+(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*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)*(d*x^3 - (2*b*d + 1)*x^2 - a^
2 + (b^2*d + 2*a)*x)), x)

Giac [F]

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

[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+(b^2*d+2*a)*x-(2*b*d+1)*x^2+d*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)*(d*x^3 - (2*b*d + 1)*x^2 - a^
2 + (b^2*d + 2*a)*x)), x)

Mupad [F(-1)]

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

[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 + b^2*d) + d*x^3 - x^2*(2*b
*d + 1) - a^2)),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 + b^2*d) + d*x^3 - x^2*(2*b
*d + 1) - a^2)), x)