\(\int \frac {(-a b+(2 a-b) x) (a^2-2 a x+x^2)}{\sqrt [3]{x (-a+x) (-b+x)} (a^4 d-4 a^3 d x+(-b^2+6 a^2 d) x^2+2 (b-2 a d) x^3+(-1+d) x^4)} \, dx\) [2984]

Optimal result

Integrand size = 91, antiderivative size = 387 \[ \int \frac {(-a b+(2 a-b) x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a \sqrt [6]{d}-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (-a \sqrt [6]{d}+\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (a \sqrt [6]{d}-\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2+\left (-a \sqrt [6]{d}+\sqrt [6]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*(2*a - b)*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x^4*(-a + x^3)^(5/3))/((-b + x^3)^
(1/3)*(a^4*d - 4*a^3*d*x^3 - b^2*(1 - (6*a^2*d)/b^2)*x^6 + 2*b*(1 - (2*a*d)/b)*x^9 - (1 - d)*x^12)), x], x, x^
(1/3)])/((a - x)*(b - x)*x)^(1/3) + (3*a*b*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][(x*(-
a + x^3)^(5/3))/((-b + x^3)^(1/3)*(-(a^4*d) + 4*a^3*d*x^3 + b^2*(1 - (6*a^2*d)/b^2)*x^6 - 2*b*(1 - (2*a*d)/b)*
x^9 + (1 - d)*x^12)), x], x, x^(1/3)])/((a - x)*(b - x)*x)^(1/3)

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a+x)^2 (-a b+(2 a-b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a d) x^3+(-1+d) x^4\right )} \, dx \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{5/3} (-a b+(2 a-b) x)}{\sqrt [3]{x} \sqrt [3]{-b+x} \left (a^4 d-4 a^3 d x+\left (-b^2+6 a^2 d\right ) x^2+2 (b-2 a 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 \left (-a+x^3\right )^{5/3} \left (-a b+(2 a-b) x^3\right )}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3+\left (-b^2+6 a^2 d\right ) x^6+2 (b-2 a 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 {(2 a-b) x^4 \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6+2 b \left (1-\frac {2 a d}{b}\right ) x^9-(1-d) x^{12}\right )}+\frac {a b x \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (-a^4 d+4 a^3 d x^3+b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6-2 b \left (1-\frac {2 a d}{b}\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^4 \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (a^4 d-4 a^3 d x^3-b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6+2 b \left (1-\frac {2 a d}{b}\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 \left (-a+x^3\right )^{5/3}}{\sqrt [3]{-b+x^3} \left (-a^4 d+4 a^3 d x^3+b^2 \left (1-\frac {6 a^2 d}{b^2}\right ) x^6-2 b \left (1-\frac {2 a d}{b}\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]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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