Integrand size = 39, antiderivative size = 52 \[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^5}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^5}}\right )}{\sqrt [4]{a}} \]
Time = 11.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [4]{a} \sqrt {x \left (-1+x^4\right )}}\right )-\text {arctanh}\left (\frac {x}{\sqrt [4]{a} \sqrt {x \left (-1+x^4\right )}}\right )}{\sqrt [4]{a}} \]
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.
\(\displaystyle \int \frac {3 x^5+x}{\sqrt {x^5-x} \left (a x^8-2 a x^4+a-x^2\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (3 x^4+1\right )}{\sqrt {x^5-x} \left (a x^8-2 a x^4+a-x^2\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x^4-1} \int \frac {\sqrt {x} \left (3 x^4+1\right )}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}dx}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^4-1} \int \frac {x \left (3 x^4+1\right )}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^4-1} \int \left (\frac {3 x^5}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}+\frac {x}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}\right )d\sqrt {x}}{\sqrt {x^5-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^4-1} \left (\int \frac {x}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}d\sqrt {x}+3 \int \frac {x^5}{\sqrt {x^4-1} \left (a x^8-2 a x^4-x^2+a\right )}d\sqrt {x}\right )}{\sqrt {x^5-x}}\) |
3.7.61.3.1 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 = 2.45 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33
method | result | size |
pseudoelliptic | \(-\frac {\left (\frac {1}{a}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{a}\right )^{\frac {1}{4}} x +\sqrt {x^{5}-x}}{-\left (\frac {1}{a}\right )^{\frac {1}{4}} x +\sqrt {x^{5}-x}}\right )+2 \arctan \left (\frac {\sqrt {x^{5}-x}}{x \left (\frac {1}{a}\right )^{\frac {1}{4}}}\right )\right )}{2}\) | \(69\) |
-1/2*(1/a)^(1/4)*(ln(((1/a)^(1/4)*x+(x^5-x)^(1/2))/(-(1/a)^(1/4)*x+(x^5-x) ^(1/2)))+2*arctan((x^5-x)^(1/2)/x/(1/a)^(1/4)))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 357, normalized size of antiderivative = 6.87 \[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=-\frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} + 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} - 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} - 2 \, \sqrt {x^{5} - x} {\left (i \, a^{\frac {1}{4}} x + \frac {-i \, a x^{4} + i \, a}{a^{\frac {1}{4}}}\right )} + a - \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} - 2 \, \sqrt {x^{5} - x} {\left (-i \, a^{\frac {1}{4}} x + \frac {i \, a x^{4} - i \, a}{a^{\frac {1}{4}}}\right )} + a - \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} \]
-1/4*log((a*x^8 - 2*a*x^4 + x^2 + 2*sqrt(x^5 - x)*(a^(1/4)*x + (a*x^4 - a) /a^(1/4)) + a + 2*(a*x^5 - a*x)/sqrt(a))/(a*x^8 - 2*a*x^4 - x^2 + a))/a^(1 /4) + 1/4*log((a*x^8 - 2*a*x^4 + x^2 - 2*sqrt(x^5 - x)*(a^(1/4)*x + (a*x^4 - a)/a^(1/4)) + a + 2*(a*x^5 - a*x)/sqrt(a))/(a*x^8 - 2*a*x^4 - x^2 + a)) /a^(1/4) - 1/4*I*log((a*x^8 - 2*a*x^4 + x^2 - 2*sqrt(x^5 - x)*(I*a^(1/4)*x + (-I*a*x^4 + I*a)/a^(1/4)) + a - 2*(a*x^5 - a*x)/sqrt(a))/(a*x^8 - 2*a*x ^4 - x^2 + a))/a^(1/4) + 1/4*I*log((a*x^8 - 2*a*x^4 + x^2 - 2*sqrt(x^5 - x )*(-I*a^(1/4)*x + (I*a*x^4 - I*a)/a^(1/4)) + a - 2*(a*x^5 - a*x)/sqrt(a))/ (a*x^8 - 2*a*x^4 - x^2 + a))/a^(1/4)
Timed out. \[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\int { \frac {3 \, x^{5} + x}{{\left (a x^{8} - 2 \, a x^{4} - x^{2} + a\right )} \sqrt {x^{5} - x}} \,d x } \]
\[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\int { \frac {3 \, x^{5} + x}{{\left (a x^{8} - 2 \, a x^{4} - x^{2} + a\right )} \sqrt {x^{5} - x}} \,d x } \]
Time = 8.75 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.98 \[ \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx=\frac {\ln \left (\frac {x-2\,a^{1/4}\,\sqrt {x^5-x}-\sqrt {a}+\sqrt {a}\,x^4}{x+\sqrt {a}-\sqrt {a}\,x^4}\right )}{2\,a^{1/4}}+\frac {\ln \left (\frac {x+\sqrt {a}-\sqrt {a}\,x^4+a^{1/4}\,\sqrt {x^5-x}\,2{}\mathrm {i}}{x-\sqrt {a}+\sqrt {a}\,x^4}\right )\,1{}\mathrm {i}}{2\,a^{1/4}} \]