Integrand size = 63, antiderivative size = 232 \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}{-2 b x+2 x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (b x-x^2+\sqrt [3]{d} \left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (b^2 x^2-2 b x^3+x^4+\left (-b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \left (a b x+(-a-b) x^2+x^3\right )^{2/3}+d^{2/3} \left (a b x+(-a-b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \] Output:
3^(1/2)*arctan(3^(1/2)*d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)/(-2*b*x+2*x^2+ d^(1/3)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)))/d^(2/3)+ln(b*x-x^2+d^(1/3)*(a*b*x+( -a-b)*x^2+x^3)^(2/3))/d^(2/3)-1/2*ln(b^2*x^2-2*b*x^3+x^4+(-b*d^(1/3)*x+d^( 1/3)*x^2)*(a*b*x+(-a-b)*x^2+x^3)^(2/3)+d^(2/3)*(a*b*x+(-a-b)*x^2+x^3)^(4/3 ))/d^(2/3)
\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \] Input:
Integrate[(a^2*b - 2*a^2*x + (2*a - b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*( a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)),x]
Output:
Integrate[(a^2*b - 2*a^2*x + (2*a - b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*( a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)), x]
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.
| \(\displaystyle \int \frac {a^2 b-2 a^2 x+x^2 (2 a-b)}{(x (x-a) (x-b))^{2/3} \left (a^2 d+x (b-2 a d)+(d-1) x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {b a^2-2 x a^2+(2 a-b) x^2}{x^{2/3} \left (x^2-(a+b) x+a b\right )^{2/3} \left (d a^2-(1-d) x^2+(b-2 a d) x\right )}dx}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \frac {b a^2-2 x a^2+(2 a-b) x^2}{\left (x^2-(a+b) x+a b\right )^{2/3} \left (d a^2-(1-d) x^2+(b-2 a d) x\right )}d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \int \left (-\frac {2 a-b}{(1-d) \left (x^2-(a+b) x+a b\right )^{2/3}}-\frac {a^2 (-2 d b+b+2 a d)-\left (2 (d+1) a^2-2 b (d+1) a+b^2\right ) x}{(d-1) \left (x^2-(a+b) x+a b\right )^{2/3} \left (d a^2+(d-1) x^2+(b-2 a d) x\right )}\right )d\sqrt [3]{x}}{(x (a-x) (b-x))^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3} \left (-\frac {\left ((2 a-b) \sqrt {4 a^2 d-4 a b d+b^2}+2 a^2 (d+1)-2 a b (d+1)+b^2\right ) \int \frac {1}{\left (b-2 a d+2 (d-1) x-\sqrt {4 d a^2-4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{1-d}-\frac {\left (-(2 a-b) \sqrt {4 a^2 d-4 a b d+b^2}+2 a^2 (d+1)-2 a b (d+1)+b^2\right ) \int \frac {1}{\left (b-2 a d+2 (d-1) x+\sqrt {4 d a^2-4 b d a+b^2}\right ) \left (x^2+(-a-b) x+a b\right )^{2/3}}d\sqrt [3]{x}}{1-d}-\frac {\sqrt [3]{x} (2 a-b) \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} \sqrt [3]{1-\frac {2 x}{-\sqrt {a^2-2 a b+b^2}+a+b}} \left (\frac {1-\frac {2 x}{\sqrt {a^2-2 a b+b^2}+a+b}}{1-\frac {2 x}{-\sqrt {a^2-2 a b+b^2}+a+b}}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {\sqrt {a^2-2 b a+b^2} x}{a b \left (1-\frac {2 x}{a+b-\sqrt {a^2-2 b a+b^2}}\right )}\right )}{(1-d) \left (1-\frac {2 x}{\sqrt {a^2-2 a b+b^2}+a+b}\right )^{2/3} \left (-x (a+b)+a b+x^2\right )^{2/3}}\right )}{(x (a-x) (b-x))^{2/3}}\) |
Input:
Int[(a^2*b - 2*a^2*x + (2*a - b)*x^2)/((x*(-a + x)*(-b + x))^(2/3)*(a^2*d + (b - 2*a*d)*x + (-1 + d)*x^2)),x]
Output:
$Aborted
\[\int \frac {a^{2} b -2 a^{2} x +\left (2 a -b \right ) x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{2} d +\left (-2 a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}d x\]
Input:
int((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a*d+b)* x+(-1+d)*x^2),x)
Output:
int((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a*d+b)* x+(-1+d)*x^2),x)
Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a *d+b)*x+(-1+d)*x^2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((a**2*b-2*a**2*x+(2*a-b)*x**2)/(x*(-a+x)*(-b+x))**(2/3)/(a**2*d+ (-2*a*d+b)*x+(-1+d)*x**2),x)
Output:
Timed out
\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \] Input:
integrate((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a *d+b)*x+(-1+d)*x^2),x, algorithm="maxima")
Output:
integrate((a^2*b - 2*a^2*x + (2*a - b)*x^2)/((a^2*d + (d - 1)*x^2 - (2*a*d - b)*x)*((a - x)*(b - x)*x)^(2/3)), x)
\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int { \frac {a^{2} b - 2 \, a^{2} x + {\left (2 \, a - b\right )} x^{2}}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}} \,d x } \] Input:
integrate((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a *d+b)*x+(-1+d)*x^2),x, algorithm="giac")
Output:
integrate((a^2*b - 2*a^2*x + (2*a - b)*x^2)/((a^2*d + (d - 1)*x^2 - (2*a*d - b)*x)*((a - x)*(b - x)*x)^(2/3)), x)
Timed out. \[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=\int \frac {x^2\,\left (2\,a-b\right )+a^2\,b-2\,a^2\,x}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (a^2\,d+x\,\left (b-2\,a\,d\right )+x^2\,\left (d-1\right )\right )} \,d x \] Input:
int((x^2*(2*a - b) + a^2*b - 2*a^2*x)/((x*(a - x)*(b - x))^(2/3)*(a^2*d + x*(b - 2*a*d) + x^2*(d - 1))),x)
Output:
int((x^2*(2*a - b) + a^2*b - 2*a^2*x)/((x*(a - x)*(b - x))^(2/3)*(a^2*d + x*(b - 2*a*d) + x^2*(d - 1))), x)
\[ \int \frac {a^2 b-2 a^2 x+(2 a-b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx=2 \left (\int \frac {x^{2}}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a^{2} d -2 x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d +x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x \right ) a -\left (\int \frac {x^{2}}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a^{2} d -2 x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d +x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x \right ) b -2 \left (\int \frac {x}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a^{2} d -2 x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d +x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x \right ) a^{2}+\left (\int \frac {1}{x^{\frac {2}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a^{2} d -2 x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} a d +x^{\frac {5}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} b +x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}} d -x^{\frac {8}{3}} \left (a b -a x -b x +x^{2}\right )^{\frac {2}{3}}}d x \right ) a^{2} b \] Input:
int((a^2*b-2*a^2*x+(2*a-b)*x^2)/(x*(-a+x)*(-b+x))^(2/3)/(a^2*d+(-2*a*d+b)* x+(-1+d)*x^2),x)
Output:
2*int(x**2/(x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*a**2*d - 2*x**(2/3)*( a*b - a*x - b*x + x**2)**(2/3)*a*d*x + x**(2/3)*(a*b - a*x - b*x + x**2)** (2/3)*b*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*d*x**2 - x**(2/3)*(a* b - a*x - b*x + x**2)**(2/3)*x**2),x)*a - int(x**2/(x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*a**2*d - 2*x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*a*d *x + x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*b*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*d*x**2 - x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*x**2) ,x)*b - 2*int(x/(x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*a**2*d - 2*x**(2 /3)*(a*b - a*x - b*x + x**2)**(2/3)*a*d*x + x**(2/3)*(a*b - a*x - b*x + x* *2)**(2/3)*b*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*d*x**2 - x**(2/3 )*(a*b - a*x - b*x + x**2)**(2/3)*x**2),x)*a**2 + int(1/(x**(2/3)*(a*b - a *x - b*x + x**2)**(2/3)*a**2*d - 2*x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3 )*a*d*x + x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)*b*x + x**(2/3)*(a*b - a *x - b*x + x**2)**(2/3)*d*x**2 - x**(2/3)*(a*b - a*x - b*x + x**2)**(2/3)* x**2),x)*a**2*b