\(\int \frac {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3)} \, dx\) [1641]

Optimal result

Integrand size = 81, antiderivative size = 111 \[ \int \frac {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{x}\right )}{d^{3/4}} \] Output:

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

Mathematica [A] (verified)

Time = 11.87 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59 \[ \int \frac {3 a b^2-2 b (2 a+b) x+(a+2 b) x^2}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-a b^2 d+b (2 a+b) d x-(a+2 b) d x^2+(-1+d) x^3\right )} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^2}}{x}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{x (-a+x) (-b+x)^2}}{x}\right )\right )}{d^{3/4}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 1387

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {-x \left (\frac {1}{d}\right )^{\frac {1}{4}}-\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}{x \left (\frac {1}{d}\right )^{\frac {1}{4}}-\left (-x \left (a -x \right ) \left (b -x \right )^{2}\right )^{\frac {1}{4}}}\right )}{\left (\frac {1}{d}\right )^{\frac {1}{4}} d}\)	\(101\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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