Integrand size = 36, antiderivative size = 45 \[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right ) \] Output:
2*(x^5+x)^(1/2)/x-2*a^(1/2)*arctanh(a^(1/2)*(x^5+x)^(1/2)/(x^4+1))
Time = 10.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2 \sqrt {x+x^5}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right ) \] Input:
Integrate[((1 + x^4)*(-1 + 3*x^4))/(x*(1 - a*x + x^4)*Sqrt[x + x^5]),x]
Output:
(2*Sqrt[x + x^5])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[x + x^5])/(1 + 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.
\(\displaystyle \int \frac {\left (x^4+1\right ) \left (3 x^4-1\right )}{x \sqrt {x^5+x} \left (-a x+x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^4+1} \int -\frac {\left (1-3 x^4\right ) \sqrt {x^4+1}}{x^{3/2} \left (x^4-a x+1\right )}dx}{\sqrt {x^5+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^4+1} \int \frac {\left (1-3 x^4\right ) \sqrt {x^4+1}}{x^{3/2} \left (x^4-a x+1\right )}dx}{\sqrt {x^5+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4+1} \int \frac {\left (1-3 x^4\right ) \sqrt {x^4+1}}{x \left (x^4-a x+1\right )}d\sqrt {x}}{\sqrt {x^5+x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4+1} \int \left (\frac {\sqrt {x^4+1} \left (4 x^3-a\right )}{-x^4+a x-1}+\frac {\sqrt {x^4+1}}{x}\right )d\sqrt {x}}{\sqrt {x^5+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4+1} \left (-a \int \frac {\sqrt {x^4+1}}{-x^4+a x-1}d\sqrt {x}-4 \int \frac {x^3 \sqrt {x^4+1}}{x^4-a x+1}d\sqrt {x}+\frac {4}{7} x^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{8},\frac {15}{8},-x^4\right )-\frac {\sqrt {x^4+1}}{\sqrt {x}}\right )}{\sqrt {x^5+x}}\) |
Input:
Int[((1 + x^4)*(-1 + 3*x^4))/(x*(1 - a*x + x^4)*Sqrt[x + x^5]),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{4}+1\right )}}{x \sqrt {a}}\right ) x +2 \sqrt {x \left (x^{4}+1\right )}}{x}\) | \(40\) |
Input:
int((x^4+1)*(3*x^4-1)/x/(x^4-a*x+1)/(x^5+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2*a^(1/2)*arctanh((x*(x^4+1))^(1/2)/x/a^(1/2))*x+2*(x*(x^4+1))^(1/2))/x
Time = 0.95 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.44 \[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\left [\frac {\sqrt {a} x \log \left (-\frac {x^{8} + 6 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 4 \, \sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{8} - 2 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 2 \, a x + 1}\right ) + 4 \, \sqrt {x^{5} + x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {\sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right ) + 2 \, \sqrt {x^{5} + x}}{x}\right ] \] Input:
integrate((x^4+1)*(3*x^4-1)/x/(x^4-a*x+1)/(x^5+x)^(1/2),x, algorithm="fric as")
Output:
[1/2*(sqrt(a)*x*log(-(x^8 + 6*a*x^5 + a^2*x^2 + 2*x^4 - 4*sqrt(x^5 + x)*(x ^4 + a*x + 1)*sqrt(a) + 6*a*x + 1)/(x^8 - 2*a*x^5 + a^2*x^2 + 2*x^4 - 2*a* x + 1)) + 4*sqrt(x^5 + x))/x, (sqrt(-a)*x*arctan(1/2*sqrt(x^5 + x)*(x^4 + a*x + 1)*sqrt(-a)/(a*x^5 + a*x)) + 2*sqrt(x^5 + x))/x]
Timed out. \[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\text {Timed out} \] Input:
integrate((x**4+1)*(3*x**4-1)/x/(x**4-a*x+1)/(x**5+x)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} {\left (x^{4} + 1\right )}}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )} x} \,d x } \] Input:
integrate((x^4+1)*(3*x^4-1)/x/(x^4-a*x+1)/(x^5+x)^(1/2),x, algorithm="maxi ma")
Output:
integrate((3*x^4 - 1)*(x^4 + 1)/(sqrt(x^5 + x)*(x^4 - a*x + 1)*x), x)
\[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} {\left (x^{4} + 1\right )}}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )} x} \,d x } \] Input:
integrate((x^4+1)*(3*x^4-1)/x/(x^4-a*x+1)/(x^5+x)^(1/2),x, algorithm="giac ")
Output:
integrate((3*x^4 - 1)*(x^4 + 1)/(sqrt(x^5 + x)*(x^4 - a*x + 1)*x), x)
Time = 7.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11 \[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=\frac {2\,\sqrt {x^5+x}}{x}+\sqrt {a}\,\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5+x}+x^4+1}{x^4-a\,x+1}\right ) \] Input:
int(((x^4 + 1)*(3*x^4 - 1))/(x*(x + x^5)^(1/2)*(x^4 - a*x + 1)),x)
Output:
(2*(x + x^5)^(1/2))/x + a^(1/2)*log((a*x - 2*a^(1/2)*(x + x^5)^(1/2) + x^4 + 1)/(x^4 - a*x + 1))
\[ \int \frac {\left (1+x^4\right ) \left (-1+3 x^4\right )}{x \left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx=-3 \left (\int \frac {\sqrt {x}\, \sqrt {x^{4}+1}\, x^{2}}{-x^{4}+a x -1}d x \right )+\int \frac {\sqrt {x}\, \sqrt {x^{4}+1}}{-x^{6}+a \,x^{3}-x^{2}}d x \] Input:
int((x^4+1)*(3*x^4-1)/x/(x^4-a*x+1)/(x^5+x)^(1/2),x)
Output:
- 3*int((sqrt(x)*sqrt(x**4 + 1)*x**2)/(a*x - x**4 - 1),x) + int((sqrt(x)* sqrt(x**4 + 1))/(a*x**3 - x**6 - x**2),x)