Integrand size = 36, antiderivative size = 28 \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-x+x^6}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]
\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx \]
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.
\(\displaystyle \int \frac {x^4 \left (4 x^5-9\right )}{\sqrt {x^6-x} \left (a x^9-x^5+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5-1} \int -\frac {x^{7/2} \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5-1} \int \frac {x^{7/2} \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}dx}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \frac {x^4 \left (9-4 x^5\right )}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \int \left (\frac {-4 x^5+9 a x^4+4}{a \sqrt {x^5-1} \left (a x^9-x^5+1\right )}-\frac {4}{a \sqrt {x^5-1}}\right )d\sqrt {x}}{\sqrt {x^6-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5-1} \left (\frac {4 \int \frac {1}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}d\sqrt {x}}{a}-\frac {4 \int \frac {x^5}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}d\sqrt {x}}{a}+9 \int \frac {x^4}{\sqrt {x^5-1} \left (a x^9-x^5+1\right )}d\sqrt {x}-\frac {4 \sqrt {x} \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},x^5\right )}{a \sqrt {x^5-1}}\right )}{\sqrt {x^6-x}}\) |
3.4.41.3.1 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]
\[\int \frac {x^{4} \left (4 x^{5}-9\right )}{\sqrt {x^{6}-x}\, \left (a \,x^{9}-x^{5}+1\right )}d x\]
Time = 0.64 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.93 \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\left [\frac {\log \left (-\frac {a^{2} x^{18} + 6 \, a x^{14} - 6 \, a x^{9} + x^{10} - 2 \, x^{5} - 4 \, {\left (a x^{13} + x^{9} - x^{4}\right )} \sqrt {x^{6} - x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{14} + 2 \, a x^{9} + x^{10} - 2 \, x^{5} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a} x^{4}}{a x^{9} + x^{5} - 1}\right )}{a}\right ] \]
[1/2*log(-(a^2*x^18 + 6*a*x^14 - 6*a*x^9 + x^10 - 2*x^5 - 4*(a*x^13 + x^9 - x^4)*sqrt(x^6 - x)*sqrt(a) + 1)/(a^2*x^18 - 2*a*x^14 + 2*a*x^9 + x^10 - 2*x^5 + 1))/sqrt(a), sqrt(-a)*arctan(2*sqrt(x^6 - x)*sqrt(-a)*x^4/(a*x^9 + x^5 - 1))/a]
\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (4 x^{5} - 9\right )}{\sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (a x^{9} - x^{5} + 1\right )}\, dx \]
Integral(x**4*(4*x**5 - 9)/(sqrt(x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))*( a*x**9 - x**5 + 1)), x)
\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} - 9\right )} x^{4}}{{\left (a x^{9} - x^{5} + 1\right )} \sqrt {x^{6} - x}} \,d x } \]
\[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} - 9\right )} x^{4}}{{\left (a x^{9} - x^{5} + 1\right )} \sqrt {x^{6} - x}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (-9+4 x^5\right )}{\sqrt {-x+x^6} \left (1-x^5+a x^9\right )} \, dx=\int \frac {x^4\,\left (4\,x^5-9\right )}{\sqrt {x^6-x}\,\left (a\,x^9-x^5+1\right )} \,d x \]