\(\int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} (b+a^2 d-(1+2 a d) x+d x^2)} \, dx\) [2468]

Optimal result

Integrand size = 55, antiderivative size = 201 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}{-2 b+2 x+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (b-x+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (b^2-2 b x+x^2+\left (-b \sqrt [3]{d}+\sqrt [3]{d} x\right ) \left (a b+(-a-b) x+x^2\right )^{2/3}+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{4/3}\right )}{2 d^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 3.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.89 \[ \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx=-\frac {(a-x)^{2/3} (b-x)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} (a-x)^{2/3}}{\sqrt [3]{d} (a-x)^{2/3}-2 \sqrt [3]{b-x}}\right )-2 \log \left (\sqrt [3]{d} (a-x)^{2/3}+\sqrt [3]{b-x}\right )+\log \left (d^{2/3} (a-x)^{4/3}-\sqrt [3]{d} (a-x)^{2/3} \sqrt [3]{b-x}+(b-x)^{2/3}\right )\right )}{2 d^{2/3} ((-a+x) (-b+x))^{2/3}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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