\(\int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{(x^2 (-a+x) (-b+x))^{2/3} (-a^2 d+2 a d x-(b+d) x^2+x^3)} \, dx\) [2661]

Optimal result

Integrand size = 73, antiderivative size = 238 \[ \int \frac {-2 a^2 b x+a (3 a+2 b) x^2-4 a x^3+x^4}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (-a^2 d+2 a d x-(b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{d}-\sqrt {3} \sqrt [3]{d} x}{a \sqrt [3]{d}-\sqrt [3]{d} x-2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(9*a*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^4*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(a
^2*d - 2*a*d*x^3 + b*(1 + d/b)*x^6 - x^9)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3) + (6*a*b*x^(4/3)*(-a
+ x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(-(a^2*d) + 2*a*d*x^3
- b*(1 + d/b)*x^6 + x^9)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3) + (3*x^(4/3)*(-a + x)^(2/3)*(-b + x)^
(2/3)*Defer[Subst][Defer[Int][(x^7*(-a + x^3)^(1/3))/((-b + x^3)^(2/3)*(-(a^2*d) + 2*a*d*x^3 - b*(1 + d/b)*x^6
+ x^9)), x], x, x^(1/3)])/((a - x)*(b - x)*x^2)^(2/3)

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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