\(\int \frac {-a b+x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} (a b d-(1+a d+b d) x+d x^2)} \, dx\) [2292]

Optimal result

Integrand size = 50, antiderivative size = 175 \[ \int \frac {-a b+x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 15.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.79 \[ \int \frac {-a b+x^2}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )-2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )+\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x +2 \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}+\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a -x \right ) \left (b -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \left (\frac {1}{d}\right )^{\frac {1}{3}} d}\)	\(145\)

[In]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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