\(\int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} (-a^2 b^2+2 a b (a+b) x-(a^2+4 a b+b^2) x^2+2 (a+b) x^3+(-1+d) x^4)} \, dx\) [2854]

Optimal result

Integrand size = 80, antiderivative size = 297 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{\sqrt [3]{d} x^2+2 \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{4 d^{2/3}} \]

[Out]

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

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

[In]

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

[Out]

(6*a*b*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^7/((-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(a
^2*b^2 - 2*a^2*b*(1 + b/a)*x^3 + a^2*(1 + (b*(4*a + b))/a^2)*x^6 - 2*a*(1 + b/a)*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^10/((-a + x^3)^(1/3)*(-b + x^3)^(1/3)*(a^2*b^2 - 2*a^2*b*(1 + b/a)*x^3 + a^2*(1 + (b*(4*a + b))/a^2)*x^6 -
2*a*(1 + b/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^{5/3} (-2 a b+(a+b) x)}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) 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^7 \left (-2 a b+(a+b) x^3\right )}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (-a^2 b^2+2 a b (a+b) x^3-\left (a^2+4 a b+b^2\right ) x^6+2 (a+b) 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^7}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+(1-d) x^{12}\right )}+\frac {(-a-b) x^{10}}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{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 (-a-b) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{a}\right ) x^9+(1-d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (6 a b \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-a+x^3} \sqrt [3]{-b+x^3} \left (a^2 b^2-2 a^2 b \left (1+\frac {b}{a}\right ) x^3+a^2 \left (1+\frac {b (4 a+b)}{a^2}\right ) x^6-2 a \left (1+\frac {b}{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 [A] (verified)

Time = 10.85 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 (-2 a b+(a+b) x)}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 (x (-a+x) (-b+x))^{2/3}}{\sqrt [3]{d} x^2}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )-2 \log \left (\sqrt [6]{d} x+\sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )+\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{x (-a+x) (-b+x)}+(x (-a+x) (-b+x))^{2/3}\right )}{4 d^{2/3}} \]

[In]

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

[Out]

(2*Sqrt[3]*ArcTan[(1 + (2*(x*(-a + x)*(-b + x))^(2/3))/(d^(1/3)*x^2))/Sqrt[3]] - 2*Log[-(d^(1/6)*x) + (x*(-a +
x)*(-b + x))^(1/3)] - 2*Log[d^(1/6)*x + (x*(-a + x)*(-b + x))^(1/3)] + Log[d^(1/3)*x^2 - d^(1/6)*x*(x*(-a + x
)*(-b + x))^(1/3) + (x*(-a + x)*(-b + x))^(2/3)] + Log[d^(1/3)*x^2 + d^(1/6)*x*(x*(-a + x)*(-b + x))^(1/3) + (
x*(-a + x)*(-b + x))^(2/3)])/(4*d^(2/3))

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.45

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x^{2}+2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{3}} x^{2}}\right )-2 \ln \left (\frac {-d^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {d^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}} x +\left (b -x \right ) \left (a -x \right ) \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}+d^{\frac {2}{3}} x^{3}}{x^{3}}\right )}{4 d^{\frac {2}{3}}}\)	\(135\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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