\(\int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{(x (-a+x) (-b+x)^2)^{2/3} (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3)} \, dx\) [2631]

Optimal result

Integrand size = 72, antiderivative size = 232 \[ \int \frac {2 a b^2 x-b (2 a+b) x^2+x^4}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-3*(b - x)*x*(1 - x/a)^(2/3)*(1 - x/b)^(1/3)*AppellF1[1/3, 2/3, 1/3, 4/3, x/a, x/b])/(-((a - x)*(b - x)^2*x))
^(2/3) + (3*b*(4*a + b)*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][x^3/((-a + x^3)^(2/3)*(-
b + x^3)^(1/3)*(a*b^2 - 2*a*b*(1 + b/(2*a))*x^3 + a*(1 + (2*b + d)/a)*x^6 - x^9)), x], x, x^(1/3)])/(-((a - x)
*(b - x)^2*x))^(2/3) - (3*(a + 3*b + d)*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][x^6/((-a
+ x^3)^(2/3)*(-b + x^3)^(1/3)*(a*b^2 - 2*a*b*(1 + b/(2*a))*x^3 + a*(1 + (2*b + d)/a)*x^6 - x^9)), x], x, x^(1
/3)])/(-((a - x)*(b - x)^2*x))^(2/3) + (3*a*b^2*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][
1/((-a + x^3)^(2/3)*(-b + x^3)^(1/3)*(-(a*b^2) + 2*a*b*(1 + b/(2*a))*x^3 - a*(1 + (2*b + d)/a)*x^6 + x^9)), x]
, x, x^(1/3)])/(-((a - x)*(b - x)^2*x))^(2/3)

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (2 a b^2-b (2 a+b) x+x^3\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{x} \left (2 a b^2-b (2 a+b) x+x^3\right )}{(-a+x)^{2/3} (-b+x)^{4/3} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{x} \left (-2 a b+b x+x^2\right )}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\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 \left (-2 a b+b x^3+x^6\right )}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\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 {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3}}+\frac {a b^2-b (4 a+b) x^3+(a+3 b+d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\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 {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3}} \, 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 {a b^2-b (4 a+b) x^3+(a+3 b+d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+b (2 a+b) x^3-(a+2 b+d) x^6+x^9\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 {b (4 a+b) x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )}+\frac {(-a-3 b-d) x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )}+\frac {a b^2}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 x^{2/3} (-b+x)^{4/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-b+x^3} \left (1-\frac {x^3}{a}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = \frac {\left (3 a b^2 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 b (4 a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 (-a-3 b-d) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 x^{2/3} (-b+x) \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^3}{a}\right )^{2/3} \sqrt [3]{1-\frac {x^3}{b}}} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ & = -\frac {3 (b-x) x \left (1-\frac {x}{a}\right )^{2/3} \sqrt [3]{1-\frac {x}{b}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},\frac {1}{3},\frac {4}{3},\frac {x}{a},\frac {x}{b}\right )}{\left (-\left ((a-x) (b-x)^2 x\right )\right )^{2/3}}+\frac {\left (3 a b^2 x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (-a b^2+2 a b \left (1+\frac {b}{2 a}\right ) x^3-a \left (1+\frac {2 b+d}{a}\right ) x^6+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 b (4 a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (3 (-a-3 b-d) x^{2/3} (-a+x)^{2/3} (-b+x)^{4/3}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^3\right )^{2/3} \sqrt [3]{-b+x^3} \left (a b^2-2 a b \left (1+\frac {b}{2 a}\right ) x^3+a \left (1+\frac {2 b+d}{a}\right ) x^6-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x (-a+x) (-b+x)^2\right )^{2/3}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 d^{\frac {2}{3}}}\)	\(134\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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