\(\int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3)} \, dx\) [2769]

Optimal result

Integrand size = 69, antiderivative size = 263 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{2 \sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}+\left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \] Output:

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

Mathematica [A] (verified)

Time = 14.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.60 \[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x (-a+x) (-b+x)^2}}{2 \sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}}\right )+2 \log \left (-\sqrt [3]{d} x+\sqrt [3]{x (-a+x) (-b+x)^2}\right )-\log \left (d^{2/3} x^2+\sqrt [3]{d} x \sqrt [3]{x (-a+x) (-b+x)^2}+\left (x (-a+x) (-b+x)^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 7293

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \int \left (\frac {\sqrt [3]{x-b} x^{7/3}}{\sqrt [3]{a-x} \left (x^3-a \left (\frac {2 b+d}{a}+1\right ) x^2+2 a b \left (\frac {b}{2 a}+1\right ) x-a b^2\right )}+\frac {b \sqrt [3]{x-b} x^{4/3}}{\sqrt [3]{a-x} \left (x^3-a \left (\frac {2 b+d}{a}+1\right ) x^2+2 a b \left (\frac {b}{2 a}+1\right ) x-a b^2\right )}+\frac {2 a b \sqrt [3]{x-b} \sqrt [3]{x}}{\sqrt [3]{a-x} \left (-x^3+a \left (\frac {2 b+d}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a-x} (x-b)^{2/3} \sqrt [3]{-a b^2-x^2 (a+2 b)+b x (2 a+b)+x^3} \left (2 a b \int \frac {\sqrt [3]{x} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (-x^3+a \left (\frac {2 b+d}{a}+1\right ) x^2-2 a b \left (\frac {b}{2 a}+1\right ) x+a b^2\right )}d\sqrt [3]{x}+b \int \frac {x^{4/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (x^3-a \left (\frac {2 b+d}{a}+1\right ) x^2+2 a b \left (\frac {b}{2 a}+1\right ) x-a b^2\right )}d\sqrt [3]{x}+\int \frac {x^{7/3} \sqrt [3]{x-b}}{\sqrt [3]{a-x} \left (x^3-a \left (\frac {2 b+d}{a}+1\right ) x^2+2 a b \left (\frac {b}{2 a}+1\right ) x-a b^2\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )} \sqrt [3]{-\left (x (a-x) (b-x)^2\right )}}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.51

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{3}} x +2 \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-d^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {d^{\frac {2}{3}} x^{2}+d^{\frac {1}{3}} \left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{3}} x +\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 d^{\frac {1}{3}}}\)	\(134\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {2 a b^2-b (2 a+b) x+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a b^2+b (2 a+b) x-(a+2 b+d) x^2+x^3\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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