Integrand size = 40, antiderivative size = 47 \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\frac {2 \sqrt {-x+x^6}}{-1+x^5}-2 \sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^6}}\right ) \]
\[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx \]
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.
\(\displaystyle \int \frac {4 x^6+x}{\left (x^5-1\right ) \sqrt {x^6-x} \left (a x^5-a-x\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (4 x^5+1\right )}{\left (x^5-1\right ) \sqrt {x^6-x} \left (a x^5-a-x\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5-1} \int -\frac {\sqrt {x} \left (4 x^5+1\right )}{\left (x^5-1\right )^{3/2} \left (-a x^5+x+a\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5-1} \int \frac {\sqrt {x} \left (4 x^5+1\right )}{\left (x^5-1\right )^{3/2} \left (-a x^5+x+a\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \frac {x \left (4 x^5+1\right )}{\left (x^5-1\right )^{3/2} \left (-a x^5+x+a\right )}d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \left (\frac {x (5 a+4 x)}{a \left (x^5-1\right )^{3/2} \left (-a x^5+x+a\right )}-\frac {4 x}{a \left (x^5-1\right )^{3/2}}\right )d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \left (-5 \int \frac {x}{\left (x^5-1\right )^{3/2} \left (a x^5-x-a\right )}d\sqrt {x}-\frac {4 \int \frac {x^2}{\left (x^5-1\right )^{3/2} \left (a x^5-x-a\right )}d\sqrt {x}}{a}+\frac {8 \sqrt {1-x^5} x^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{10},\frac {1}{2},\frac {13}{10},x^5\right )}{15 a \sqrt {x^5-1}}+\frac {4 x^{3/2}}{5 a \sqrt {x^5-1}}\right )}{\sqrt {x^6-x}}\) |
3.7.1.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = 45, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}-x}\, \sqrt {a}}{x}\right ) \sqrt {x^{6}-x}+2 x}{\sqrt {x^{6}-x}}\) | \(45\) |
Time = 0.35 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.91 \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\left [\frac {{\left (x^{5} - 1\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} - 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} - a + x\right )} \sqrt {x^{6} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{10} - 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} + 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{6} - x}}{2 \, {\left (x^{5} - 1\right )}}, \frac {{\left (x^{5} - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{a x^{5} - a + x}\right ) + 2 \, \sqrt {x^{6} - x}}{x^{5} - 1}\right ] \]
[1/2*((x^5 - 1)*sqrt(a)*log(-(a^2*x^10 - 2*a^2*x^5 + 6*a*x^6 - 4*(a*x^5 - a + x)*sqrt(x^6 - x)*sqrt(a) + a^2 - 6*a*x + x^2)/(a^2*x^10 - 2*a^2*x^5 - 2*a*x^6 + a^2 + 2*a*x + x^2)) + 4*sqrt(x^6 - x))/(x^5 - 1), ((x^5 - 1)*sqr t(-a)*arctan(2*sqrt(x^6 - x)*sqrt(-a)/(a*x^5 - a + x)) + 2*sqrt(x^6 - x))/ (x^5 - 1)]
Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\text {Timed out} \]
\[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {4 \, x^{6} + x}{{\left (a x^{5} - a - x\right )} \sqrt {x^{6} - x} {\left (x^{5} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=-\int \frac {4\,x^6+x}{\sqrt {x^6-x}\,\left (x^5-1\right )\,\left (-a\,x^5+x+a\right )} \,d x \]