Integrand size = 36, antiderivative size = 28 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]
\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \]
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.
\(\displaystyle \int \frac {x^4 \left (5 x^4-9\right )}{\sqrt {x^5-x} \left (a x^9-x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^4-1} \int -\frac {x^{7/2} \left (9-5 x^4\right )}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}dx}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^4-1} \int \frac {x^{7/2} \left (9-5 x^4\right )}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}dx}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \int \frac {x^4 \left (9-5 x^4\right )}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \int \left (\frac {9 x^4}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}-\frac {5 x^8}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}\right )d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^4-1} \left (9 \int \frac {x^4}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}d\sqrt {x}-5 \int \frac {x^8}{\sqrt {x^4-1} \left (a x^9-x^4+1\right )}d\sqrt {x}\right )}{\sqrt {x^5-x}}\) |
3.4.40.3.1 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]
\[\int \frac {x^{4} \left (5 x^{4}-9\right )}{\sqrt {x^{5}-x}\, \left (a \,x^{9}-x^{4}+1\right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.36 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{18} + 6 \, a x^{13} - 6 \, a x^{9} + x^{8} - 2 \, x^{4} - 4 \, {\left (a x^{13} + x^{8} - x^{4}\right )} \sqrt {x^{5} - x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{13} + 2 \, a x^{9} + x^{8} - 2 \, x^{4} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{9} + x^{4} - 1\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{9} - a x^{5}\right )}}\right )}{a}\right ] \]
[1/2*log((a^2*x^18 + 6*a*x^13 - 6*a*x^9 + x^8 - 2*x^4 - 4*(a*x^13 + x^8 - x^4)*sqrt(x^5 - x)*sqrt(a) + 1)/(a^2*x^18 - 2*a*x^13 + 2*a*x^9 + x^8 - 2*x ^4 + 1))/sqrt(a), sqrt(-a)*arctan(1/2*(a*x^9 + x^4 - 1)*sqrt(x^5 - x)*sqrt (-a)/(a*x^9 - a*x^5))/a]
\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (5 x^{4} - 9\right )}{\sqrt {x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (a x^{9} - x^{4} + 1\right )}\, dx \]
\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]
\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]
Time = 6.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\frac {\ln \left (\frac {a\,x^9+x^4-2\,\sqrt {a}\,x^4\,\sqrt {x\,\left (x^4-1\right )}-1}{4\,a\,x^9-4\,x^4+4}\right )}{\sqrt {a}} \]