Integrand size = 35, antiderivative size = 26 \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x^5}{\sqrt {a} \sqrt {x+x^6}}\right )}{\sqrt {a}} \]
\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx \]
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.
\(\displaystyle \int \frac {x^4 \left (4 x^5+9\right )}{\sqrt {x^6+x} \left (-a x^5-a+x^9\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5+1} \int -\frac {x^{7/2} \left (4 x^5+9\right )}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5+1} \int \frac {x^{7/2} \left (4 x^5+9\right )}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {x^4 \left (4 x^5+9\right )}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \left (\frac {4 a x^5+9 x^4+4 a}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}-\frac {4}{\sqrt {x^5+1}}\right )d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \left (4 a \int \frac {1}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}d\sqrt {x}+4 a \int \frac {x^5}{\sqrt {x^5+1} \left (-x^9+a x^5+a\right )}d\sqrt {x}-9 \int \frac {x^4}{\sqrt {x^5+1} \left (x^9-a x^5-a\right )}d\sqrt {x}-4 \sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},-x^5\right )\right )}{\sqrt {x^6+x}}\) |
3.3.95.3.1 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]
\[\int \frac {x^{4} \left (4 x^{5}+9\right )}{\sqrt {x^{6}+x}\, \left (x^{9}-a \,x^{5}-a \right )}d x\]
Time = 0.62 (sec) , antiderivative size = 144, normalized size of antiderivative = 5.54 \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\left [\frac {\log \left (-\frac {x^{18} + 6 \, a x^{14} + a^{2} x^{10} + 6 \, a x^{9} + 2 \, a^{2} x^{5} - 4 \, {\left (x^{13} + a x^{9} + a x^{4}\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2}}{x^{18} - 2 \, a x^{14} + a^{2} x^{10} - 2 \, a x^{9} + 2 \, a^{2} x^{5} + a^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a} x^{4}}{x^{9} + a x^{5} + a}\right )}{a}\right ] \]
[1/2*log(-(x^18 + 6*a*x^14 + a^2*x^10 + 6*a*x^9 + 2*a^2*x^5 - 4*(x^13 + a* x^9 + a*x^4)*sqrt(x^6 + x)*sqrt(a) + a^2)/(x^18 - 2*a*x^14 + a^2*x^10 - 2* a*x^9 + 2*a^2*x^5 + a^2))/sqrt(a), sqrt(-a)*arctan(2*sqrt(x^6 + x)*sqrt(-a )*x^4/(x^9 + a*x^5 + a))/a]
\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (4 x^{5} + 9\right )}{\sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (- a x^{5} - a + x^{9}\right )}\, dx \]
Integral(x**4*(4*x**5 + 9)/(sqrt(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))*( -a*x**5 - a + x**9)), x)
\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (x^{9} - a x^{5} - a\right )} \sqrt {x^{6} + x}} \,d x } \]
\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (x^{9} - a x^{5} - a\right )} \sqrt {x^{6} + x}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-a-a x^5+x^9\right )} \, dx=\int -\frac {x^4\,\left (4\,x^5+9\right )}{\sqrt {x^6+x}\,\left (-x^9+a\,x^5+a\right )} \,d x \]