\(\int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} (a b d-(a+b) d x+(-1+d) x^2)} \, dx\) [2234]

Optimal result

Integrand size = 48, antiderivative size = 167 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

((a + b + Sqrt[2*a*b*(2 - d) + a^2*d + b^2*d]/Sqrt[d])*x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Int][1/(x^(
1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*(-((a + b)*d) - Sqrt[d]*Sqrt[4*a*b + a^2*d - 2*a*b*d + b^2*d] + 2*(-1 + d)*
x)), x])/((a - x)*(b - x)*x)^(1/3) + ((a + b - Sqrt[2*a*b*(2 - d) + a^2*d + b^2*d]/Sqrt[d])*x^(1/3)*(-a + x)^(
1/3)*(-b + x)^(1/3)*Defer[Int][1/(x^(1/3)*(-a + x)^(1/3)*(-b + x)^(1/3)*(-((a + b)*d) + Sqrt[d]*Sqrt[4*a*b + a
^2*d - 2*a*b*d + b^2*d] + 2*(-1 + d)*x)), x])/((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 {-2 a b+(a+b) x}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {a+b+\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}+\frac {a+b-\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ & = \frac {\left (\left (a+b-\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (\left (a+b+\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.36 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.78 \[ \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}}\right )-2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x (-a+x) (-b+x)}\right )+\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)}+d^{2/3} (x (-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\)	\(137\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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