\(\int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} (b+(-1+a^2 d) x-2 a d x^2+d x^3)} \, dx\) [2551]

Optimal result

Integrand size = 75, antiderivative size = 215 \[ \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right )}{d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a^2 b^3 x+\left (3 a^2 b^2+2 a b^3\right ) x^2+\left (-3 a^2 b-6 a b^2-b^3\right ) x^3+\left (a^2+6 a b+3 b^2\right ) x^4+(-2 a-3 b) x^5+x^6}}{b-x}\right )}{d^{3/4}} \] Output:

-2*arctan(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^ 
3)*x^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(1/4)/(b-x))/d^(3/4)+2*ar 
ctanh(d^(1/4)*(-a^2*b^3*x+(3*a^2*b^2+2*a*b^3)*x^2+(-3*a^2*b-6*a*b^2-b^3)*x 
^3+(a^2+6*a*b+3*b^2)*x^4+(-2*a-3*b)*x^5+x^6)^(1/4)/(b-x))/d^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 11.66 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int \frac {-a^2 b+4 a b x-(2 a+3 b) x^2+2 x^3}{\sqrt [4]{x (-a+x)^2 (-b+x)^3} \left (b+\left (-1+a^2 d\right ) x-2 a d x^2+d x^3\right )} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x)^2 (-b+x)^3}}{b-x}\right )+\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{(a-x)^2 x (-b+x)^3}}{-b+x}\right )\right )}{d^{3/4}} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int -\frac {-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b}{\sqrt [4]{x} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}dx}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b}{\sqrt [4]{x} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}dx}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {\sqrt {x} \left (-2 x^3+(2 a+3 b) x^2-4 a b x+a^2 b\right )}{\left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right ) \sqrt [4]{x^5-(2 a+3 b) x^4+\left (a^2+6 b a+3 b^2\right ) x^3-b \left (3 a^2+6 b a+b^2\right ) x^2+a b^2 (3 a+2 b) x-a^2 b^3}}d\sqrt [4]{x}}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {(a-x) \sqrt {x} \left (2 x^2-3 b x+a b\right )}{\sqrt [4]{(a-x)^2 (x-b)^3} \left (b+x \left (d a^2-2 d x a+d x^2-1\right )\right )}d\sqrt [4]{x}}{\sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 2058

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \frac {\sqrt {a-x} \sqrt {x} \left (2 x^2-3 b x+a b\right )}{(x-b)^{3/4} \left (b-x \left (-d a^2+2 d x a-d x^2+1\right )\right )}d\sqrt [4]{x}}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \int \left (\frac {2 \sqrt {a-x} x^{5/2}}{(x-b)^{3/4} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right )}+\frac {3 b \sqrt {a-x} x^{3/2}}{(x-b)^{3/4} \left (-d x^3+2 a d x^2+\left (1-a^2 d\right ) x-b\right )}+\frac {a b \sqrt {a-x} \sqrt {x}}{(x-b)^{3/4} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt {a-x} (x-b)^{3/4} \sqrt [4]{-a^2 b^3+x^3 \left (a^2+6 a b+3 b^2\right )-b x^2 \left (3 a^2+6 a b+b^2\right )+a b^2 x (3 a+2 b)-x^4 (2 a+3 b)+x^5} \left (a b \int \frac {\sqrt {a-x} \sqrt {x}}{(x-b)^{3/4} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right )}d\sqrt [4]{x}+3 b \int \frac {\sqrt {a-x} x^{3/2}}{(x-b)^{3/4} \left (-d x^3+2 a d x^2+\left (1-a^2 d\right ) x-b\right )}d\sqrt [4]{x}+2 \int \frac {\sqrt {a-x} x^{5/2}}{(x-b)^{3/4} \left (d x^3-2 a d x^2-\left (1-a^2 d\right ) x+b\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{-(a-x)^2 (b-x)^3} \sqrt [4]{x \left (-(a-x)^2\right ) (b-x)^3}}\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

2*int(x**3/(x**(1/4)*( - a**2*b**3 + 3*a**2*b**2*x - 3*a**2*b*x**2 + a**2* 
x**3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x**3 - 2*a*x**4 - b**3*x**2 + 3* 
b**2*x**3 - 3*b*x**4 + x**5)**(1/4)*a**2*d*x - 2*x**(1/4)*( - a**2*b**3 + 
3*a**2*b**2*x - 3*a**2*b*x**2 + a**2*x**3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6 
*a*b*x**3 - 2*a*x**4 - b**3*x**2 + 3*b**2*x**3 - 3*b*x**4 + x**5)**(1/4)*a 
*d*x**2 + x**(1/4)*( - a**2*b**3 + 3*a**2*b**2*x - 3*a**2*b*x**2 + a**2*x* 
*3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x**3 - 2*a*x**4 - b**3*x**2 + 3*b* 
*2*x**3 - 3*b*x**4 + x**5)**(1/4)*b + x**(1/4)*( - a**2*b**3 + 3*a**2*b**2 
*x - 3*a**2*b*x**2 + a**2*x**3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x**3 - 
 2*a*x**4 - b**3*x**2 + 3*b**2*x**3 - 3*b*x**4 + x**5)**(1/4)*d*x**3 - x** 
(1/4)*( - a**2*b**3 + 3*a**2*b**2*x - 3*a**2*b*x**2 + a**2*x**3 + 2*a*b**3 
*x - 6*a*b**2*x**2 + 6*a*b*x**3 - 2*a*x**4 - b**3*x**2 + 3*b**2*x**3 - 3*b 
*x**4 + x**5)**(1/4)*x),x) - 2*int(x**2/(x**(1/4)*( - a**2*b**3 + 3*a**2*b 
**2*x - 3*a**2*b*x**2 + a**2*x**3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x** 
3 - 2*a*x**4 - b**3*x**2 + 3*b**2*x**3 - 3*b*x**4 + x**5)**(1/4)*a**2*d*x 
- 2*x**(1/4)*( - a**2*b**3 + 3*a**2*b**2*x - 3*a**2*b*x**2 + a**2*x**3 + 2 
*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x**3 - 2*a*x**4 - b**3*x**2 + 3*b**2*x** 
3 - 3*b*x**4 + x**5)**(1/4)*a*d*x**2 + x**(1/4)*( - a**2*b**3 + 3*a**2*b** 
2*x - 3*a**2*b*x**2 + a**2*x**3 + 2*a*b**3*x - 6*a*b**2*x**2 + 6*a*b*x**3 
- 2*a*x**4 - b**3*x**2 + 3*b**2*x**3 - 3*b*x**4 + x**5)**(1/4)*b + x**(...