\(\int \frac {x (4 a b-3 (a+b) x+2 x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a b d+(a+b) d x-d x^2+x^4)} \, dx\) [2515]

Optimal result

Integrand size = 59, antiderivative size = 210 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 x^2+\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{d^{2/3}}+\frac {\log \left (x^2-\sqrt [3]{d} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{d^{2/3}}-\frac {\log \left (x^4+\sqrt [3]{d} x^2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+d^{2/3} \left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 d^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 12.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}{2 x^2+\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (x^2-\sqrt [3]{d} \sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (x^4+\sqrt [3]{d} x^2 \sqrt [3]{x^2 (-a+x) (-b+x)}+d^{2/3} \left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 d^{2/3}} \] Input:

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

Output:

(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(x^2*(-a + x)*(-b + x))^(1/3))/(2*x^2 + 
 d^(1/3)*(x^2*(-a + x)*(-b + x))^(1/3))] + 2*Log[x^2 - d^(1/3)*(x^2*(-a + 
x)*(-b + x))^(1/3)] - Log[x^4 + d^(1/3)*x^2*(x^2*(-a + x)*(-b + x))^(1/3) 
+ d^(2/3)*(x^2*(-a + x)*(-b + x))^(2/3)])/(2*d^(2/3))
 

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

int(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x- 
d*x^2+x^4),x)
 

Output:

int(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x- 
d*x^2+x^4),x)
 

Fricas [F(-1)]

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

integrate(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b 
)*d*x-d*x^2+x^4),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(x*(4*a*b-3*(a+b)*x+2*x**2)/(x**2*(-a+x)*(-b+x))**(1/3)/(-a*b*d+( 
a+b)*d*x-d*x**2+x**4),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b 
)*d*x-d*x^2+x^4),x, algorithm="maxima")
 

Output:

integrate((4*a*b - 3*(a + b)*x + 2*x^2)*x/(((a - x)*(b - x)*x^2)^(1/3)*(x^ 
4 - a*b*d + (a + b)*d*x - d*x^2)), x)
 

Giac [F]

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

integrate(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b 
)*d*x-d*x^2+x^4),x, algorithm="giac")
 

Output:

integrate((4*a*b - 3*(a + b)*x + 2*x^2)*x/(((a - x)*(b - x)*x^2)^(1/3)*(x^ 
4 - a*b*d + (a + b)*d*x - d*x^2)), x)
 

Mupad [F(-1)]

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

int(-(x*(4*a*b + 2*x^2 - 3*x*(a + b)))/((x^2*(a - x)*(b - x))^(1/3)*(d*x^2 
 - x^4 - d*x*(a + b) + a*b*d)),x)
 

Output:

-int((x*(4*a*b + 2*x^2 - 3*x*(a + b)))/((x^2*(a - x)*(b - x))^(1/3)*(d*x^2 
 - x^4 - d*x*(a + b) + a*b*d)), x)
 

Reduce [F]

\[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b d+(a+b) d x-d x^2+x^4\right )} \, dx=-2 \left (\int \frac {x^{3}}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a b d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} b d +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} d -x^{\frac {14}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}}}d x \right )+3 \left (\int \frac {x^{2}}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a b d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} b d +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} d -x^{\frac {14}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}}}d x \right ) a +3 \left (\int \frac {x^{2}}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a b d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} b d +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} d -x^{\frac {14}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}}}d x \right ) b -4 \left (\int \frac {x}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a b d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} a d -x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} b d +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}} d -x^{\frac {14}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {1}{3}}}d x \right ) a b \] Input:

int(x*(4*a*b-3*(a+b)*x+2*x^2)/(x^2*(-a+x)*(-b+x))^(1/3)/(-a*b*d+(a+b)*d*x- 
d*x^2+x^4),x)
 

Output:

 - 2*int(x**3/(x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*a*b*d - x**(2/3)*( 
a*b - a*x - b*x + x**2)**(1/3)*a*d*x - x**(2/3)*(a*b - a*x - b*x + x**2)** 
(1/3)*b*d*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*d*x**2 - x**(2/3)*( 
a*b - a*x - b*x + x**2)**(1/3)*x**4),x) + 3*int(x**2/(x**(2/3)*(a*b - a*x 
- b*x + x**2)**(1/3)*a*b*d - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*a*d* 
x - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*b*d*x + x**(2/3)*(a*b - a*x - 
 b*x + x**2)**(1/3)*d*x**2 - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*x**4 
),x)*a + 3*int(x**2/(x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*a*b*d - x**( 
2/3)*(a*b - a*x - b*x + x**2)**(1/3)*a*d*x - x**(2/3)*(a*b - a*x - b*x + x 
**2)**(1/3)*b*d*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*d*x**2 - x**( 
2/3)*(a*b - a*x - b*x + x**2)**(1/3)*x**4),x)*b - 4*int(x/(x**(2/3)*(a*b - 
 a*x - b*x + x**2)**(1/3)*a*b*d - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3) 
*a*d*x - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3)*b*d*x + x**(2/3)*(a*b - 
a*x - b*x + x**2)**(1/3)*d*x**2 - x**(2/3)*(a*b - a*x - b*x + x**2)**(1/3) 
*x**4),x)*a*b