Integrand size = 25, antiderivative size = 17 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=2 \text {arctanh}\left (\frac {(-2+x) x}{\sqrt {x+x^4}}\right ) \] Output:
2*arctanh((-2+x)*x/(x^4+x)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(17)=34\).
Time = 8.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {2 \sqrt {x} \sqrt {1+x^3} \text {arctanh}\left (\frac {(-2+x) \sqrt {x}}{\sqrt {1+x^3}}\right )}{\sqrt {x+x^4}} \] Input:
Integrate[(2 + x + 2*x^2)/((-1 + 2*x)*Sqrt[x + x^4]),x]
Output:
(2*Sqrt[x]*Sqrt[1 + x^3]*ArcTanh[((-2 + x)*Sqrt[x])/Sqrt[1 + x^3]])/Sqrt[x + x^4]
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.
| \(\displaystyle \int \frac {2 x^2+x+2}{(2 x-1) \sqrt {x^4+x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^3+1} \int -\frac {2 x^2+x+2}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^3+1} \int \frac {2 x^2+x+2}{(1-2 x) \sqrt {x} \sqrt {x^3+1}}dx}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \frac {2 x^2+x+2}{(1-2 x) \sqrt {x^3+1}}d\sqrt {x}}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \int \left (-\frac {x}{\sqrt {x^3+1}}+\frac {3}{(1-2 x) \sqrt {x^3+1}}-\frac {1}{\sqrt {x^3+1}}\right )d\sqrt {x}}{\sqrt {x^4+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^3+1} \left (\frac {3}{2} \int \frac {1}{\left (1-\sqrt {2} \sqrt {x}\right ) \sqrt {x^3+1}}d\sqrt {x}+\frac {3}{2} \int \frac {1}{\left (\sqrt {2} \sqrt {x}+1\right ) \sqrt {x^3+1}}d\sqrt {x}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{3} \text {arcsinh}\left (x^{3/2}\right )\right )}{\sqrt {x^4+x}}\) |
Input:
Int[(2 + x + 2*x^2)/((-1 + 2*x)*Sqrt[x + x^4]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(15)=30\).
Time = 0.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.65
| method | result | size |
| trager | \(\ln \left (-\frac {2 x^{3}+2 x \sqrt {x^{4}+x}-4 x^{2}-4 \sqrt {x^{4}+x}+4 x +1}{\left (-1+2 x \right )^{2}}\right )\) | \(45\) |
| default | \(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(512\) |
| elliptic | \(\text {Expression too large to display}\) | \(781\) |
Input:
int((2*x^2+x+2)/(-1+2*x)/(x^4+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
ln(-(2*x^3+2*x*(x^4+x)^(1/2)-4*x^2-4*(x^4+x)^(1/2)+4*x+1)/(-1+2*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.12 \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\frac {1}{3} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \frac {2}{3} \, \log \left (-\frac {10 \, x^{3} - 6 \, x^{2} - 6 \, \sqrt {x^{4} + x} {\left (x + 1\right )} + 12 \, x + 1}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}\right ) \] Input:
integrate((2*x^2+x+2)/(-1+2*x)/(x^4+x)^(1/2),x, algorithm="fricas")
Output:
1/3*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + 2/3*log(-(10*x^3 - 6*x^2 - 6*sqr t(x^4 + x)*(x + 1) + 12*x + 1)/(8*x^3 - 12*x^2 + 6*x - 1))
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {2 x^{2} + x + 2}{\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x - 1\right )}\, dx \] Input:
integrate((2*x**2+x+2)/(-1+2*x)/(x**4+x)**(1/2),x)
Output:
Integral((2*x**2 + x + 2)/(sqrt(x*(x + 1)*(x**2 - x + 1))*(2*x - 1)), x)
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {2 \, x^{2} + x + 2}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \] Input:
integrate((2*x^2+x+2)/(-1+2*x)/(x^4+x)^(1/2),x, algorithm="maxima")
Output:
integrate((2*x^2 + x + 2)/(sqrt(x^4 + x)*(2*x - 1)), x)
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int { \frac {2 \, x^{2} + x + 2}{\sqrt {x^{4} + x} {\left (2 \, x - 1\right )}} \,d x } \] Input:
integrate((2*x^2+x+2)/(-1+2*x)/(x^4+x)^(1/2),x, algorithm="giac")
Output:
integrate((2*x^2 + x + 2)/(sqrt(x^4 + x)*(2*x - 1)), x)
Timed out. \[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=\int \frac {2\,x^2+x+2}{\left (2\,x-1\right )\,\sqrt {x^4+x}} \,d x \] Input:
int((x + 2*x^2 + 2)/((2*x - 1)*(x + x^4)^(1/2)),x)
Output:
int((x + 2*x^2 + 2)/((2*x - 1)*(x + x^4)^(1/2)), x)
\[ \int \frac {2+x+2 x^2}{(-1+2 x) \sqrt {x+x^4}} \, dx=2 \left (\int \frac {\sqrt {x}\, \sqrt {x^{3}+1}\, x}{2 x^{4}-x^{3}+2 x -1}d x \right )+2 \left (\int \frac {\sqrt {x}\, \sqrt {x^{3}+1}}{2 x^{5}-x^{4}+2 x^{2}-x}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{3}+1}}{2 x^{4}-x^{3}+2 x -1}d x \] Input:
int((2*x^2+x+2)/(-1+2*x)/(x^4+x)^(1/2),x)
Output:
2*int((sqrt(x)*sqrt(x**3 + 1)*x)/(2*x**4 - x**3 + 2*x - 1),x) + 2*int((sqr t(x)*sqrt(x**3 + 1))/(2*x**5 - x**4 + 2*x**2 - x),x) + int((sqrt(x)*sqrt(x **3 + 1))/(2*x**4 - x**3 + 2*x - 1),x)