\(\int \frac {x (-a b+x^2)}{(x^2 (-a+x) (-b+x))^{2/3} (a b d-(1+a d+b d) x+d x^2)} \, dx\) [2284]

Optimal result

Integrand size = 51, antiderivative size = 174 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

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

Rubi [F]

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

[In]

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

[Out]

(3*x^2*(1 - x/a)^(2/3)*(1 - x/b)^(2/3)*AppellF1[2/3, 2/3, 2/3, 5/3, x/a, x/b])/(2*d*((a - x)*(b - x)*x^2)^(2/3
)) + ((1 + a*d + b*d + Sqrt[a^2*d^2 + 2*a*d*(1 - b*d) + (1 + b*d)^2])*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*De
fer[Int][1/(x^(1/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*(-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)^(2/3)) + ((1 + a*d + b*d - Sqrt[a^2*d^2 + 2*a*d*(1 - b*d
) + (1 + b*d)^2])*x^(4/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*Defer[Int][1/(x^(1/3)*(-a + x)^(2/3)*(-b + x)^(2/3)*(-
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
)^(2/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {-a b+x^2}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b d-(1+a d+b d) x+d x^2\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 \left (\frac {1}{d \sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3}}-\frac {2 a b d-(1+a d+b d) x}{d \sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b d+(-1-a d-b d) x+d x^2\right )}\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 {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3}} \, dx}{d \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 b d-(1+a d+b d) x}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b d+(-1-a d-b d) x+d x^2\right )} \, dx}{d \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 \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}}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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}}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (x^{4/3} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}} \, dx}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}} \\ & = -\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^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\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^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (x^{4/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}} \, dx}{d \left (x^2 (-a+x) (-b+x)\right )^{2/3}} \\ & = \frac {3 x^2 \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {2}{3},\frac {2}{3},\frac {5}{3},\frac {x}{a},\frac {x}{b}\right )}{2 d \left ((a-x) (b-x) x^2\right )^{2/3}}-\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^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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 \left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\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^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \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 \left (x^2 (-a+x) (-b+x)\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.42 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[In]

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

[Out]

(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)]
)/(2*d^(1/3))

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.83

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

[In]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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