\(\int \frac {x (-a+x) (-b+x) (a b-2 b x+x^2)}{(x (-a+x) (-b+x)^2)^{2/3} (-b^2+2 b x-(1-a^2 d) x^2-2 a d x^3+d x^4)} \, dx\) [2928]

Optimal result

Integrand size = 78, antiderivative size = 340 \[ \int \frac {x (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}}\right )}{2 d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}+d^{2/3} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{4/3}\right )}{4 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 15.69 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.62 \[ \int \frac {x (-a+x) (-b+x) \left (a b-2 b x+x^2\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+2 b x-\left (1-a^2 d\right ) x^2-2 a d x^3+d x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}}\right )-\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\sqrt [3]{d} (b-x)^2 \left ((b-x)^2 x (-a+x)\right )^{2/3}+d^{2/3} \left ((b-x)^2 x (-a+x)\right )^{4/3}\right )+2 \log \left (b^2-2 b x+x^2-\sqrt [3]{d} \left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{4 d^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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