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

Optimal result

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

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)
 

Mathematica [A] (verified)

Time = 15.41 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 \sqrt [3]{d}} \] Input:

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]
 

Output:

(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))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 x^{4/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \left (-\frac {(a d+b d+1) \left (-\sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}+a d+b d+1\right ) \int \frac {\sqrt [3]{x}}{\left (a d+b d-2 x d-\sqrt {a^2 d^2+b^2 d^2-2 a b d^2+2 a d+2 b d+1}+1\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{d \sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}}+\frac {(a d+b d+1) \left (\sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}+a d+b d+1\right ) \int \frac {\sqrt [3]{x}}{\left (a d+b d-2 x d+\sqrt {a^2 d^2+b^2 d^2-2 a b d^2+2 a d+2 b d+1}+1\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{d \sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}}-\frac {4 a b d \int \frac {\sqrt [3]{x}}{\left (a d+b d-2 x d+\sqrt {a^2 d^2+b^2 d^2-2 a b d^2+2 a d+2 b d+1}+1\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}}-\frac {4 a b d \int \frac {\sqrt [3]{x}}{\left (-a d-b d+2 x d+\sqrt {a^2 d^2+b^2 d^2-2 a b d^2+2 a d+2 b d+1}-1\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{\sqrt {a^2 d^2+2 a d (1-b d)+(b d+1)^2}}-\frac {x^{2/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {2}{3},\frac {2}{3},\frac {5}{3},\frac {2 x}{a+b+\sqrt {a^2-2 b a+b^2}},\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}{2 d \left (-x (a+b)+a b+x^2\right )^{2/3}}\right )}{\left (x^2 (a-x) (b-x)\right )^{2/3}}\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.83

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

Input:

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,m 
ethod=_RETURNVERBOSE)
 

Output:

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

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")
 

Output:

Timed out
 

Sympy [F(-1)]

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

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)
 

Output:

Timed out
 

Maxima [F]

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

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")
 

Output:

-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]

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

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")
 

Output:

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

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)
                                                                                    
                                                                                    
 

Output:

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)
 

Reduce [F]

\[ \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b d-(1+a d+b d) x+d x^2\right )} \, dx=\int \frac {x^{2}}{x^{\frac {1}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a b d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b d +x^{\frac {7}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x -\left (\int \frac {1}{x^{\frac {1}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a b d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b d +x^{\frac {7}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {4}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x \right ) a b \] Input:

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)
 

Output:

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