Integrand size = 37, antiderivative size = 52 \[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^6}}\right )}{a^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^6}}\right )}{a^{3/4}} \] Output:
arctan(a^(1/4)*x/(x^6-x)^(1/2))/a^(3/4)-arctanh(a^(1/4)*x/(x^6-x)^(1/2))/a ^(3/4)
\[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx \] Input:
Integrate[(x + 4*x^6)/(Sqrt[-x + x^6]*(1 - a*x^2 - 2*x^5 + x^10)),x]
Output:
Integrate[(x + 4*x^6)/(Sqrt[-x + x^6]*(1 - a*x^2 - 2*x^5 + x^10)), 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.
\(\displaystyle \int \frac {4 x^6+x}{\sqrt {x^6-x} \left (-a x^2+x^{10}-2 x^5+1\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (4 x^5+1\right )}{\sqrt {x^6-x} \left (-a x^2+x^{10}-2 x^5+1\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5-1} \int \frac {\sqrt {x} \left (4 x^5+1\right )}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5-1} \int \frac {x \left (4 x^5+1\right )}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5-1} \int \left (\frac {4 x^6}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}+\frac {x}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5-1} \left (\int \frac {x}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}d\sqrt {x}+4 \int \frac {x^6}{\sqrt {x^5-1} \left (x^{10}-2 x^5-a x^2+1\right )}d\sqrt {x}\right )}{\sqrt {x^6-x}}\) |
Input:
Int[(x + 4*x^6)/(Sqrt[-x + x^6]*(1 - a*x^2 - 2*x^5 + x^10)),x]
Output:
$Aborted
Time = 1.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.17
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {a^{\frac {1}{4}} x +\sqrt {x^{6}-x}}{-a^{\frac {1}{4}} x +\sqrt {x^{6}-x}}\right )+2 \arctan \left (\frac {\sqrt {x^{6}-x}}{a^{\frac {1}{4}} x}\right )}{2 a^{\frac {3}{4}}}\) | \(61\) |
Input:
int((4*x^6+x)/(x^6-x)^(1/2)/(x^10-2*x^5-a*x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/2/a^(3/4)*(ln((a^(1/4)*x+(x^6-x)^(1/2))/(-a^(1/4)*x+(x^6-x)^(1/2)))+2*a rctan(1/a^(1/4)/x*(x^6-x)^(1/2)))
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 429, normalized size of antiderivative = 8.25 \[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=-\frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} x^{10} - 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + 2 \, \sqrt {x^{6} - x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x - 1\right )} + 2 \, {\left (a x^{6} - a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )}}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} x^{10} - 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} - 2 \, \sqrt {x^{6} - x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x - 1\right )} + 2 \, {\left (a x^{6} - a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )}}\right ) - \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (i \, a^{2} x^{10} - 2 i \, a^{2} x^{5} + i \, a^{3} x^{2} + i \, a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} - 2 \, \sqrt {x^{6} - x} {\left (x^{5} - a^{2} \sqrt {\frac {1}{a^{3}}} x - 1\right )} + 2 \, {\left (-i \, a x^{6} + i \, a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )}}\right ) + \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (-i \, a^{2} x^{10} + 2 i \, a^{2} x^{5} - i \, a^{3} x^{2} - i \, a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} - 2 \, \sqrt {x^{6} - x} {\left (x^{5} - a^{2} \sqrt {\frac {1}{a^{3}}} x - 1\right )} + 2 \, {\left (i \, a x^{6} - i \, a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )}}\right ) \] Input:
integrate((4*x^6+x)/(x^6-x)^(1/2)/(x^10-2*x^5-a*x^2+1),x, algorithm="frica s")
Output:
-1/4*(a^(-3))^(1/4)*log(1/2*((a^2*x^10 - 2*a^2*x^5 + a^3*x^2 + a^2)*(a^(-3 ))^(3/4) + 2*sqrt(x^6 - x)*(x^5 + a^2*sqrt(a^(-3))*x - 1) + 2*(a*x^6 - a*x )*(a^(-3))^(1/4))/(x^10 - 2*x^5 - a*x^2 + 1)) + 1/4*(a^(-3))^(1/4)*log(-1/ 2*((a^2*x^10 - 2*a^2*x^5 + a^3*x^2 + a^2)*(a^(-3))^(3/4) - 2*sqrt(x^6 - x) *(x^5 + a^2*sqrt(a^(-3))*x - 1) + 2*(a*x^6 - a*x)*(a^(-3))^(1/4))/(x^10 - 2*x^5 - a*x^2 + 1)) - 1/4*I*(a^(-3))^(1/4)*log(-1/2*((I*a^2*x^10 - 2*I*a^2 *x^5 + I*a^3*x^2 + I*a^2)*(a^(-3))^(3/4) - 2*sqrt(x^6 - x)*(x^5 - a^2*sqrt (a^(-3))*x - 1) + 2*(-I*a*x^6 + I*a*x)*(a^(-3))^(1/4))/(x^10 - 2*x^5 - a*x ^2 + 1)) + 1/4*I*(a^(-3))^(1/4)*log(-1/2*((-I*a^2*x^10 + 2*I*a^2*x^5 - I*a ^3*x^2 - I*a^2)*(a^(-3))^(3/4) - 2*sqrt(x^6 - x)*(x^5 - a^2*sqrt(a^(-3))*x - 1) + 2*(I*a*x^6 - I*a*x)*(a^(-3))^(1/4))/(x^10 - 2*x^5 - a*x^2 + 1))
\[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\int \frac {x \left (4 x^{5} + 1\right )}{\sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (- a x^{2} + x^{10} - 2 x^{5} + 1\right )}\, dx \] Input:
integrate((4*x**6+x)/(x**6-x)**(1/2)/(x**10-2*x**5-a*x**2+1),x)
Output:
Integral(x*(4*x**5 + 1)/(sqrt(x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))*(-a* x**2 + x**10 - 2*x**5 + 1)), x)
\[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\int { \frac {4 \, x^{6} + x}{{\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )} \sqrt {x^{6} - x}} \,d x } \] Input:
integrate((4*x^6+x)/(x^6-x)^(1/2)/(x^10-2*x^5-a*x^2+1),x, algorithm="maxim a")
Output:
integrate((4*x^6 + x)/((x^10 - 2*x^5 - a*x^2 + 1)*sqrt(x^6 - x)), x)
\[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\int { \frac {4 \, x^{6} + x}{{\left (x^{10} - 2 \, x^{5} - a x^{2} + 1\right )} \sqrt {x^{6} - x}} \,d x } \] Input:
integrate((4*x^6+x)/(x^6-x)^(1/2)/(x^10-2*x^5-a*x^2+1),x, algorithm="giac" )
Output:
integrate((4*x^6 + x)/((x^10 - 2*x^5 - a*x^2 + 1)*sqrt(x^6 - x)), x)
Timed out. \[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=\int -\frac {4\,x^6+x}{\sqrt {x^6-x}\,\left (-x^{10}+2\,x^5+a\,x^2-1\right )} \,d x \] Input:
int(-(x + 4*x^6)/((x^6 - x)^(1/2)*(a*x^2 + 2*x^5 - x^10 - 1)),x)
Output:
int(-(x + 4*x^6)/((x^6 - x)^(1/2)*(a*x^2 + 2*x^5 - x^10 - 1)), x)
\[ \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (1-a x^2-2 x^5+x^{10}\right )} \, dx=-4 \left (\int \frac {\sqrt {x}\, \sqrt {x^{5}-1}\, x^{5}}{-x^{15}+3 x^{10}+a \,x^{7}-3 x^{5}-a \,x^{2}+1}d x \right )-\left (\int \frac {\sqrt {x}\, \sqrt {x^{5}-1}}{-x^{15}+3 x^{10}+a \,x^{7}-3 x^{5}-a \,x^{2}+1}d x \right ) \] Input:
int((4*x^6+x)/(x^6-x)^(1/2)/(x^10-2*x^5-a*x^2+1),x)
Output:
- 4*int((sqrt(x)*sqrt(x**5 - 1)*x**5)/(a*x**7 - a*x**2 - x**15 + 3*x**10 - 3*x**5 + 1),x) - int((sqrt(x)*sqrt(x**5 - 1))/(a*x**7 - a*x**2 - x**15 + 3*x**10 - 3*x**5 + 1),x)