\(\int \frac {x (4 a b-3 (a+b) x+2 x^2)}{\sqrt [3]{x^2 (-a+x) (-b+x)} (-a b+(a+b) x-x^2+d x^4)} \, dx\) [2481]

Optimal result

Integrand size = 58, antiderivative size = 204 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a b x^2+(-a-b) x^3+x^4}}{2 \sqrt [3]{d} x^2+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d} x^2+\sqrt [3]{a b x^2+(-a-b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^4+\sqrt [3]{d} x^2 \sqrt [3]{a b x^2+(-a-b) x^3+x^4}+\left (a b x^2+(-a-b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \] Output:

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

Mathematica [A] (verified)

Time = 12.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (4 a b-3 (a+b) x+2 x^2\right )}{\sqrt [3]{x^2 (-a+x) (-b+x)} \left (-a b+(a+b) x-x^2+d x^4\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a+x) (-b+x)}}{2 \sqrt [3]{d} x^2+\sqrt [3]{x^2 (-a+x) (-b+x)}}\right )+2 \log \left (-\sqrt [3]{d} x^2+\sqrt [3]{x^2 (-a+x) (-b+x)}\right )-\log \left (d^{2/3} x^4+\sqrt [3]{d} x^2 \sqrt [3]{x^2 (-a+x) (-b+x)}+\left (x^2 (-a+x) (-b+x)\right )^{2/3}\right )}{2 \sqrt [3]{d}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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