Integrand size = 28, antiderivative size = 24 \[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {x+x^6}}\right )}{\sqrt {a}} \]
\[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx \]
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.
\(\displaystyle \int \frac {4 x^5-1}{\sqrt {x^6+x} \left (-a x+x^5+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5+1} \int -\frac {1-4 x^5}{\sqrt {x} \sqrt {x^5+1} \left (x^5-a x+1\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5+1} \int \frac {1-4 x^5}{\sqrt {x} \sqrt {x^5+1} \left (x^5-a x+1\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {1-4 x^5}{\sqrt {x^5+1} \left (x^5-a x+1\right )}d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \left (\frac {5-4 a x}{\sqrt {x^5+1} \left (x^5-a x+1\right )}-\frac {4}{\sqrt {x^5+1}}\right )d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \left (-5 \int \frac {1}{\left (-x^5+a x-1\right ) \sqrt {x^5+1}}d\sqrt {x}+4 a \int \frac {x}{\left (-x^5+a x-1\right ) \sqrt {x^5+1}}d\sqrt {x}-4 \sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},-x^5\right )\right )}{\sqrt {x^6+x}}\) |
3.3.48.3.1 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 = 1.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{6}+x}}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(21\) |
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.96 \[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {\log \left (-\frac {x^{10} + 6 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} + x} {\left (x^{5} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{10} - 2 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{x^{5} + a x + 1}\right )}{a}\right ] \]
[1/2*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))/s qrt(a), sqrt(-a)*arctan(2*sqrt(x^6 + x)*sqrt(-a)/(x^5 + a*x + 1))/a]
\[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {4 x^{5} - 1}{\sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (- a x + x^{5} + 1\right )}\, dx \]
\[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int { \frac {4 \, x^{5} - 1}{\sqrt {x^{6} + x} {\left (x^{5} - a x + 1\right )}} \,d x } \]
\[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int { \frac {4 \, x^{5} - 1}{\sqrt {x^{6} + x} {\left (x^{5} - a x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {4\,x^5-1}{\sqrt {x^6+x}\,\left (x^5-a\,x+1\right )} \,d x \]