Integrand size = 27, antiderivative size = 54 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-3+a} x \sqrt {-1+3 x-a x^2+x^3}}{1-3 x+a x^2-x^3}\right )}{\sqrt {-3+a}} \] Output:
2*arctan((-3+a)^(1/2)*x*(-a*x^2+x^3+3*x-1)^(1/2)/(a*x^2-x^3-3*x+1))/(-3+a) ^(1/2)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3+a} x}{\sqrt {-1+3 x-a x^2+x^3}}\right )}{\sqrt {-3+a}} \] Input:
Integrate[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x - a*x^2 + x^3]),x]
Output:
(-2*ArcTan[(Sqrt[-3 + a]*x)/Sqrt[-1 + 3*x - a*x^2 + x^3]])/Sqrt[-3 + a]
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.
\(\displaystyle \int \frac {x+2}{(x-1) \sqrt {-a x^2+x^3+3 x-1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3}{(x-1) \sqrt {-a x^2+x^3+3 x-1}}+\frac {1}{\sqrt {-a x^2+x^3+3 x-1}}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {3}{(x-1) \sqrt {-a x^2+x^3+3 x-1}}+\frac {1}{\sqrt {-a x^2+x^3+3 x-1}}\right )dx\) |
Input:
Int[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x - a*x^2 + x^3]),x]
Output:
$Aborted
Time = 1.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {-3+a}}\right )}{\sqrt {-3+a}}\) | \(34\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}}{x \sqrt {-3+a}}\right )}{\sqrt {-3+a}}\) | \(34\) |
elliptic | \(\text {Expression too large to display}\) | \(3008\) |
Input:
int((2+x)/(-1+x)/(-a*x^2+x^3+3*x-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/(-3+a)^(1/2)*arctan((-a*x^2+x^3+3*x-1)^(1/2)/x/(-3+a)^(1/2))
Time = 0.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.33 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\left [-\frac {\sqrt {-a + 3} \log \left (-\frac {2 \, {\left (4 \, a - 9\right )} x^{5} - x^{6} - {\left (8 \, a^{2} - 24 \, a + 15\right )} x^{4} + 4 \, {\left (6 \, a - 13\right )} x^{3} - {\left (8 \, a - 9\right )} x^{2} - 4 \, {\left ({\left (2 \, a - 3\right )} x^{3} - x^{4} - 3 \, x^{2} + x\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {-a + 3} + 6 \, x - 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right )}{2 \, {\left (a - 3\right )}}, \frac {\arctan \left (-\frac {{\left ({\left (2 \, a - 3\right )} x^{2} - x^{3} - 3 \, x + 1\right )} \sqrt {-a x^{2} + x^{3} + 3 \, x - 1} \sqrt {a - 3}}{2 \, {\left ({\left (a - 3\right )} x^{4} - {\left (a^{2} - 3 \, a\right )} x^{3} + 3 \, {\left (a - 3\right )} x^{2} - {\left (a - 3\right )} x\right )}}\right )}{\sqrt {a - 3}}\right ] \] Input:
integrate((2+x)/(-1+x)/(-a*x^2+x^3+3*x-1)^(1/2),x, algorithm="fricas")
Output:
[-1/2*sqrt(-a + 3)*log(-(2*(4*a - 9)*x^5 - x^6 - (8*a^2 - 24*a + 15)*x^4 + 4*(6*a - 13)*x^3 - (8*a - 9)*x^2 - 4*((2*a - 3)*x^3 - x^4 - 3*x^2 + x)*sq rt(-a*x^2 + x^3 + 3*x - 1)*sqrt(-a + 3) + 6*x - 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1))/(a - 3), arctan(-1/2*((2*a - 3)*x^2 - x^3 - 3 *x + 1)*sqrt(-a*x^2 + x^3 + 3*x - 1)*sqrt(a - 3)/((a - 3)*x^4 - (a^2 - 3*a )*x^3 + 3*(a - 3)*x^2 - (a - 3)*x))/sqrt(a - 3)]
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {- a x^{2} + x^{3} + 3 x - 1}}\, dx \] Input:
integrate((2+x)/(-1+x)/(-a*x**2+x**3+3*x-1)**(1/2),x)
Output:
Integral((x + 2)/((x - 1)*sqrt(-a*x**2 + x**3 + 3*x - 1)), x)
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \] Input:
integrate((2+x)/(-1+x)/(-a*x^2+x^3+3*x-1)^(1/2),x, algorithm="maxima")
Output:
integrate((x + 2)/(sqrt(-a*x^2 + x^3 + 3*x - 1)*(x - 1)), x)
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {-a x^{2} + x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \] Input:
integrate((2+x)/(-1+x)/(-a*x^2+x^3+3*x-1)^(1/2),x, algorithm="giac")
Output:
integrate((x + 2)/(sqrt(-a*x^2 + x^3 + 3*x - 1)*(x - 1)), x)
Timed out. \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=\int \frac {x+2}{\left (x-1\right )\,\sqrt {x^3-a\,x^2+3\,x-1}} \,d x \] Input:
int((x + 2)/((x - 1)*(3*x - a*x^2 + x^3 - 1)^(1/2)),x)
Output:
int((x + 2)/((x - 1)*(3*x - a*x^2 + x^3 - 1)^(1/2)), x)
\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x-a x^2+x^3}} \, dx=-2 \left (\int \frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}}{a \,x^{3}-x^{4}-a \,x^{2}+x^{3}-3 x^{2}+4 x -1}d x \right )-\left (\int \frac {\sqrt {-a \,x^{2}+x^{3}+3 x -1}\, x}{a \,x^{3}-x^{4}-a \,x^{2}+x^{3}-3 x^{2}+4 x -1}d x \right ) \] Input:
int((2+x)/(-1+x)/(-a*x^2+x^3+3*x-1)^(1/2),x)
Output:
- 2*int(sqrt( - a*x**2 + x**3 + 3*x - 1)/(a*x**3 - a*x**2 - x**4 + x**3 - 3*x**2 + 4*x - 1),x) - int((sqrt( - a*x**2 + x**3 + 3*x - 1)*x)/(a*x**3 - a*x**2 - x**4 + x**3 - 3*x**2 + 4*x - 1),x)