\(\int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} (-a^2 b^2+2 a b (a+b) x-(a^2+4 a b+b^2) x^2+2 (a+b) x^3+(-1+d) x^4)} \, dx\) [2657]

Optimal result

Integrand size = 80, antiderivative size = 237 \[ \int \frac {x^3 (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (-a^2 b^2+2 a b (a+b) x-\left (a^2+4 a b+b^2\right ) x^2+2 (a+b) x^3+(-1+d) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{2 d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}}{\sqrt [3]{d} x^2+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{2 d^{5/6}} \]

[Out]

1/2*3^(1/2)*arctan(3^(1/2)*d^(1/6)*x/(d^(1/6)*x-2*(a*b*x+(-a-b)*x^2+x^3)^(1/3)))/d^(5/6)-1/2*3^(1/2)*arctan(3^
(1/2)*d^(1/6)*x/(d^(1/6)*x+2*(a*b*x+(-a-b)*x^2+x^3)^(1/3)))/d^(5/6)+arctanh(d^(1/6)*x/(a*b*x+(-a-b)*x^2+x^3)^(
1/3))/d^(5/6)+1/2*arctanh(d^(1/6)*x*(a*b*x+(-a-b)*x^2+x^3)^(1/3)/(d^(1/3)*x^2+(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d
^(5/6)

Rubi [F]

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

[In]

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

(-3*(a + b)*((b*(a - x))/(a*(b - x)))^(2/3)*(b - x)*x*Hypergeometric2F1[1/3, 2/3, 4/3, -(((a - b)*x)/(a*(b - x
)))])/(b*(1 - d)*((a - x)*(b - x)*x)^(2/3)) + (3*a^2*b^2*(a + b)*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[S
ubst][Defer[Int][1/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(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)])/((1 - d)*((a - x)*(b - x)*x)^(2/3)) - (6*a*b
*(a + b)^2*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^3/((-a + x^3)^(2/3)*(-b + x^3)^(2/3
)*(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)])/((1 - d)*((a - x)*(b - x)*x)^(2/3)) + (3*(a + b)*(a^2 + 4*a*b + b^2)*x^(2/3)*(-a + x)^(2/3)*(-b
+ x)^(2/3)*Defer[Subst][Defer[Int][x^6/((-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(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)])/((1 - d)*((a - x)*(b - x
)*x)^(2/3)) - (6*(a^2 + b^2 + a*b*(1 + d))*x^(2/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][x^9/(
(-a + x^3)^(2/3)*(-b + x^3)^(2/3)*(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)])/((1 - d)*((a - x)*(b - x)*x)^(2/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {x^{7/3} (-2 a b+(a+b) x)}{(-a+x)^{2/3} (-b+x)^{2/3} \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}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^9 \left (-2 a b+(a+b) x^3\right )}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(x (-a+x) (-b+x))^{2/3}} \\ & = \frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \left (-\frac {a+b}{(1-d) \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}}+\frac {a^2 b^2 (a+b)-2 a b (a+b)^2 x^3+(a+b) \left (a^2+4 a b+b^2\right ) x^6-2 \left (a^2+b^2+a b (1+d)\right ) x^9}{(-1+d) \left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(x (-a+x) (-b+x))^{2/3}} \\ & = -\frac {\left (3 (a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{(1-d) (x (-a+x) (-b+x))^{2/3}}+\frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {a^2 b^2 (a+b)-2 a b (a+b)^2 x^3+(a+b) \left (a^2+4 a b+b^2\right ) x^6-2 \left (a^2+b^2+a b (1+d)\right ) x^9}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}} \\ & = -\frac {3 (a+b) \left (\frac {b (a-x)}{a (b-x)}\right )^{2/3} (b-x) x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) ((a-x) (b-x) x)^{2/3}}+\frac {\left (3 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \left (\frac {a^2 (-a-b) b^2}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 {2 a b (a+b)^2 x^3}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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) \left (-a^2-4 a b-b^2\right ) x^6}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 {2 \left (a^2+b^2+a b (1+d)\right ) x^9}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}} \\ & = -\frac {3 (a+b) \left (\frac {b (a-x)}{a (b-x)}\right )^{2/3} (b-x) x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {(a-b) x}{a (b-x)}\right )}{b (1-d) ((a-x) (b-x) x)^{2/3}}-\frac {\left (3 a^2 b^2 (a+b) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}}+\frac {\left (6 a b (a+b)^2 x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}}-\frac {\left (3 (a+b) \left (a^2+4 a b+b^2\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}}+\frac {\left (6 \left (a^2+b^2+a b (1+d)\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\left (-a+x^3\right )^{2/3} \left (-b+x^3\right )^{2/3} \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 )}{(-1+d) (x (-a+x) (-b+x))^{2/3}} \\ \end{align*}

Mathematica [F]

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

[In]

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

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

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x -2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 x \,d^{\frac {1}{6}}}\right )+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 x \,d^{\frac {1}{6}}}\right )-\ln \left (\frac {d^{\frac {1}{6}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x -d^{\frac {1}{3}} x^{2}-\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {d^{\frac {1}{6}} x -\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+2 \ln \left (\frac {d^{\frac {1}{6}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {1}{6}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +d^{\frac {1}{3}} x^{2}+\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 d^{\frac {5}{6}}}\)	\(238\)

[In]

int(x^3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))^(2/3)/(-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^(1/6)*x-2*(x*(a-x)*(b-x))^(1/3))/x/d^(1/6))+2*3^(1/2)*arctan(1/3*3^(1/2)
*(d^(1/6)*x+2*(x*(a-x)*(b-x))^(1/3))/x/d^(1/6))-ln((d^(1/6)*(x*(a-x)*(b-x))^(1/3)*x-d^(1/3)*x^2-(x*(a-x)*(b-x)
)^(2/3))/x^2)-2*ln((d^(1/6)*x-(x*(a-x)*(b-x))^(1/3))/x)+2*ln((d^(1/6)*x+(x*(a-x)*(b-x))^(1/3))/x)+ln((d^(1/6)*
(x*(a-x)*(b-x))^(1/3)*x+d^(1/3)*x^2+(x*(a-x)*(b-x))^(2/3))/x^2))/d^(5/6)

Fricas [F(-1)]

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

[In]

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

[In]

integrate(x**3*(-2*a*b+(a+b)*x)/(x*(-a+x)*(-b+x))**(2/3)/(-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]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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