Integrand size = 36, antiderivative size = 39 \[ \int \frac {\left (1+x^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\frac {2 \sqrt {x+x^6}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {x+x^6}}\right ) \]
Time = 10.80 (sec) , antiderivative size = 39, 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 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5+1} \int -\frac {\left (1-4 x^5\right ) \sqrt {x^5+1}}{x^{3/2} \left (x^5-a x+1\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5+1} \int \frac {\left (1-4 x^5\right ) \sqrt {x^5+1}}{x^{3/2} \left (x^5-a x+1\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {\left (1-4 x^5\right ) \sqrt {x^5+1}}{x \left (x^5-a x+1\right )}d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 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 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \left (-a \int \frac {\sqrt {x^5+1}}{-x^5+a x-1}d\sqrt {x}-5 \int \frac {x^4 \sqrt {x^5+1}}{x^5-a x+1}d\sqrt {x}+\frac {5}{9} x^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{10},\frac {19}{10},-x^5\right )-\frac {\sqrt {x^5+1}}{\sqrt {x}}\right )}{\sqrt {x^6+x}}\) |
3.6.11.3.1 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 = 1.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}+x}}{x \sqrt {a}}\right ) x +2 \sqrt {x^{6}+x}}{x}\) | \(36\) |
Time = 2.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.74 \[ \int \frac {\left (1+x^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {\sqrt {a} x \log \left (-\frac {x^{10} + 6 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} + x} {\left (x^{5} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{10} - 2 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {x^{6} + x}}{2 \, x}, \frac {\sqrt {-a} x \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{x^{5} + a x + 1}\right ) + 2 \, \sqrt {x^{6} + x}}{x}\right ] \]
[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. \[ \int \frac {\left (1+x^5\right ) \left (-1+4 x^5\right )}{x \left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {\left (x^5+1\right )\,\left (4\,x^5-1\right )}{x\,\sqrt {x^6+x}\,\left (x^5-a\,x+1\right )} \,d x \]