Integrand size = 38, antiderivative size = 43 \[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\frac {2 \sqrt {-x+x^6}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {-x+x^6}}\right ) \]
Time = 10.62 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\frac {2 \sqrt {-x+x^6}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {-x+x^6}}\right ) \]
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.
\(\displaystyle \int \frac {\left (x^5-1\right ) \left (4 x^5+1\right )}{x \sqrt {x^6-x} \left (-a x+x^5-1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5-1} \int -\frac {\sqrt {x^5-1} \left (4 x^5+1\right )}{x^{3/2} \left (-x^5+a x+1\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5-1} \int \frac {\sqrt {x^5-1} \left (4 x^5+1\right )}{x^{3/2} \left (-x^5+a x+1\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \frac {\sqrt {x^5-1} \left (4 x^5+1\right )}{x \left (-x^5+a x+1\right )}d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \left (\frac {\sqrt {x^5-1} \left (5 x^4-a\right )}{-x^5+a x+1}+\frac {\sqrt {x^5-1}}{x}\right )d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \left (-a \int \frac {\sqrt {x^5-1}}{-x^5+a x+1}d\sqrt {x}-5 \int \frac {x^4 \sqrt {x^5-1}}{x^5-a x-1}d\sqrt {x}+\frac {5 \sqrt {1-x^5} x^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{10},\frac {19}{10},x^5\right )}{9 \sqrt {x^5-1}}-\frac {\sqrt {x^5-1}}{\sqrt {x}}\right )}{\sqrt {x^6-x}}\) |
3.6.58.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-2 x \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}-x}}{x \sqrt {a}}\right )+2 \sqrt {x^{6}-x}}{x}\) | \(40\) |
Time = 2.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 3.58 \[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\left [\frac {\sqrt {a} x \log \left (-\frac {x^{10} + 6 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} - x} {\left (x^{5} + a x - 1\right )} \sqrt {a} - 6 \, a x + 1}{x^{10} - 2 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} + 2 \, a x + 1}\right ) + 4 \, \sqrt {x^{6} - x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{x^{5} + a x - 1}\right ) + 2 \, \sqrt {x^{6} - x}}{x}\right ] \]
[1/2*(sqrt(a)*x*log(-(x^10 + 6*a*x^6 - 2*x^5 + a^2*x^2 - 4*sqrt(x^6 - x)*( x^5 + a*x - 1)*sqrt(a) - 6*a*x + 1)/(x^10 - 2*a*x^6 - 2*x^5 + a^2*x^2 + 2* a*x + 1)) + 4*sqrt(x^6 - x))/x, (sqrt(-a)*x*arctan(2*sqrt(x^6 - x)*sqrt(-a )/(x^5 + a*x - 1)) + 2*sqrt(x^6 - x))/x]
\[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (4 x^{5} + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}{x \sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (- a x + x^{5} - 1\right )}\, dx \]
Integral((x - 1)*(4*x**5 + 1)*(x**4 + x**3 + x**2 + x + 1)/(x*sqrt(x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))*(-a*x + x**5 - 1)), x)
\[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {{\left (4 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )} x} \,d x } \]
\[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {{\left (4 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )} x} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^5\right ) \left (1+4 x^5\right )}{x \left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=-\int \frac {\left (x^5-1\right )\,\left (4\,x^5+1\right )}{x\,\sqrt {x^6-x}\,\left (-x^5+a\,x+1\right )} \,d x \]