Integrand size = 39, antiderivative size = 56 \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\frac {2 \left (3 a+x+3 a x^5\right ) \sqrt {x+x^6}}{3 \left (1+x^5\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right ) \]
Time = 22.79 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\frac {2 \sqrt {x+x^6} \left (x+3 a \left (1+x^5\right )\right )}{3 \left (1+x^5\right )^2}-2 a^{3/2} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right ) \]
(2*Sqrt[x + x^6]*(x + 3*a*(1 + x^5)))/(3*(1 + x^5)^2) - 2*a^(3/2)*ArcTanh[ x/(Sqrt[a]*Sqrt[x + x^6])]
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.
\(\displaystyle \int \frac {x^2 \left (4 x^5-1\right )}{\left (x^5+1\right )^2 \sqrt {x^6+x} \left (a x^5+a-x\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^5+1} \int -\frac {x^{3/2} \left (1-4 x^5\right )}{\left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {x^5+1} \int \frac {x^{3/2} \left (1-4 x^5\right )}{\left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}dx}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {x^2 \left (1-4 x^5\right )}{\left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \int \left (\frac {(5 a-4 x) x^2}{a \left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}-\frac {4 x^2}{a \left (x^5+1\right )^{5/2}}\right )d\sqrt {x}}{\sqrt {x^6+x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^5+1} \left (-\frac {4 \int \frac {x^3}{\left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}d\sqrt {x}}{a}+5 \int \frac {x^2}{\left (x^5+1\right )^{5/2} \left (a x^5-x+a\right )}d\sqrt {x}-\frac {8 x^{15/2}}{15 a \left (x^5+1\right )^{3/2}}-\frac {4 x^{5/2}}{5 a \left (x^5+1\right )^{3/2}}\right )}{\sqrt {x^6+x}}\) |
3.8.28.3.1 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 = 1.58 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16
method | result | size |
pseudoelliptic | \(\frac {6 \left (-x^{5}-1\right ) a^{\frac {3}{2}} \sqrt {x^{6}+x}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}+x}\, \sqrt {a}}{x}\right )+6 x \left (a \,x^{5}+a +\frac {1}{3} x \right )}{\sqrt {x^{6}+x}\, \left (3 x^{5}+3\right )}\) | \(65\) |
6/(x^6+x)^(1/2)*((-x^5-1)*a^(3/2)*(x^6+x)^(1/2)*arctanh((x^6+x)^(1/2)/x*a^ (1/2))+x*(a*x^5+a+1/3*x))/(3*x^5+3)
Time = 0.37 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.98 \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {3 \, {\left (a x^{10} + 2 \, a x^{5} + a\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right ) + 4 \, {\left (3 \, a x^{5} + 3 \, a + x\right )} \sqrt {x^{6} + x}}{6 \, {\left (x^{10} + 2 \, x^{5} + 1\right )}}, \frac {3 \, {\left (a x^{10} + 2 \, a x^{5} + a\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{a x^{5} + a + x}\right ) + 2 \, {\left (3 \, a x^{5} + 3 \, a + x\right )} \sqrt {x^{6} + x}}{3 \, {\left (x^{10} + 2 \, x^{5} + 1\right )}}\right ] \]
[1/6*(3*(a*x^10 + 2*a*x^5 + a)*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*(3*a*x^5 + 3*a + x)*sqrt( x^6 + x))/(x^10 + 2*x^5 + 1), 1/3*(3*(a*x^10 + 2*a*x^5 + a)*sqrt(-a)*arcta n(2*sqrt(x^6 + x)*sqrt(-a)/(a*x^5 + a + x)) + 2*(3*a*x^5 + 3*a + x)*sqrt(x ^6 + x))/(x^10 + 2*x^5 + 1)]
Timed out. \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\int { \frac {{\left (4 \, x^{5} - 1\right )} x^{2}}{{\left (a x^{5} + a - x\right )} \sqrt {x^{6} + x} {\left (x^{5} + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2 \left (-1+4 x^5\right )}{\left (1+x^5\right )^2 \left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\int \frac {x^2\,\left (4\,x^5-1\right )}{{\left (x^5+1\right )}^2\,\sqrt {x^6+x}\,\left (a\,x^5-x+a\right )} \,d x \]