Integrand size = 34, antiderivative size = 199 \[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=-\frac {\sqrt {2} \left (-4+\sqrt {-4+a} \sqrt {a}+a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2-\sqrt {-4+a} \sqrt {a}-a}}\right )}{5 \sqrt {2-\sqrt {-4+a} \sqrt {a}-a} \sqrt {-4+a} \sqrt {a}}-\frac {\sqrt {2} \left (4+\sqrt {-4+a} \sqrt {a}-a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2+\sqrt {-4+a} \sqrt {a}-a}}\right )}{5 \sqrt {2+\sqrt {-4+a} \sqrt {a}-a} \sqrt {-4+a} \sqrt {a}} \]
-1/5*2^(1/2)*(-4+(-4+a)^(1/2)*a^(1/2)+a)*arctan(2^(1/2)*(x^5-1)^(1/2)/(2-( -4+a)^(1/2)*a^(1/2)-a)^(1/2))/(2-(-4+a)^(1/2)*a^(1/2)-a)^(1/2)/(-4+a)^(1/2 )/a^(1/2)-1/5*2^(1/2)*(4+(-4+a)^(1/2)*a^(1/2)-a)*arctan(2^(1/2)*(x^5-1)^(1 /2)/(2+(-4+a)^(1/2)*a^(1/2)-a)^(1/2))/(2+(-4+a)^(1/2)*a^(1/2)-a)^(1/2)/(-4 +a)^(1/2)/a^(1/2)
Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.90 \[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=-\frac {\sqrt {2} \left (\frac {\left (-4+\sqrt {-4+a} \sqrt {a}+a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2-\sqrt {-4+a} \sqrt {a}-a}}\right )}{\sqrt {2-\sqrt {-4+a} \sqrt {a}-a}}+\frac {\left (4+\sqrt {-4+a} \sqrt {a}-a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2+\sqrt {-4+a} \sqrt {a}-a}}\right )}{\sqrt {2+\sqrt {-4+a} \sqrt {a}-a}}\right )}{5 \sqrt {-4+a} \sqrt {a}} \]
-1/5*(Sqrt[2]*(((-4 + Sqrt[-4 + a]*Sqrt[a] + a)*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a]])/Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a] + ((4 + Sqrt[-4 + a]*Sqrt[a] - a)*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt [2 + Sqrt[-4 + a]*Sqrt[a] - a]])/Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]))/(Sqr t[-4 + a]*Sqrt[a])
Time = 0.59 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.33, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2027, 7266, 1197, 1478, 25, 1103}
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.
\(\displaystyle \int \frac {2 x^4-x^9}{\sqrt {x^5-1} \left (-a x^5+a+x^{10}\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^4 \left (2-x^5\right )}{\sqrt {x^5-1} \left (-a x^5+a+x^{10}\right )}dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {1}{5} \int \frac {2-x^5}{\sqrt {x^5-1} \left (x^{10}-a x^5+a\right )}dx^5\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle \frac {2}{5} \int \frac {1-x^{10}}{x^{20}+(2-a) x^{10}+1}d\sqrt {x^5-1}\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle \frac {2}{5} \left (-\frac {\int -\frac {\sqrt {a}-2 \sqrt {x^5-1}}{x^{10}-\sqrt {a} \sqrt {x^5-1}+1}d\sqrt {x^5-1}}{2 \sqrt {a}}-\frac {\int -\frac {\sqrt {a}+2 \sqrt {x^5-1}}{x^{10}+\sqrt {a} \sqrt {x^5-1}+1}d\sqrt {x^5-1}}{2 \sqrt {a}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{5} \left (\frac {\int \frac {\sqrt {a}-2 \sqrt {x^5-1}}{x^{10}-\sqrt {a} \sqrt {x^5-1}+1}d\sqrt {x^5-1}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}+2 \sqrt {x^5-1}}{x^{10}+\sqrt {a} \sqrt {x^5-1}+1}d\sqrt {x^5-1}}{2 \sqrt {a}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {2}{5} \left (\frac {\log \left (\sqrt {a} \sqrt {x^5-1}+x^{10}+1\right )}{2 \sqrt {a}}-\frac {\log \left (-\sqrt {a} \sqrt {x^5-1}+x^{10}+1\right )}{2 \sqrt {a}}\right )\) |
(2*(-1/2*Log[1 + x^10 - Sqrt[a]*Sqrt[-1 + x^5]]/Sqrt[a] + Log[1 + x^10 + S qrt[a]*Sqrt[-1 + x^5]]/(2*Sqrt[a])))/5
3.25.56.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
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[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Time = 0.77 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 \left (-4+\sqrt {a \left (-4+a \right )}+a \right ) \arctan \left (\frac {2 \sqrt {x^{5}-1}}{\sqrt {-2 \sqrt {a \left (-4+a \right )}-2 a +4}}\right )}{5 \sqrt {-2 \sqrt {a \left (-4+a \right )}-2 a +4}}-\frac {2 \left (4+\sqrt {a \left (-4+a \right )}-a \right ) \arctan \left (\frac {2 \sqrt {x^{5}-1}}{\sqrt {2 \sqrt {a \left (-4+a \right )}-2 a +4}}\right )}{5 \sqrt {2 \sqrt {a \left (-4+a \right )}-2 a +4}}}{\sqrt {a \left (-4+a \right )}}\) | \(121\) |
2/5/(a*(-4+a))^(1/2)*(-(-4+(a*(-4+a))^(1/2)+a)/(-2*(a*(-4+a))^(1/2)-2*a+4) ^(1/2)*arctan(2*(x^5-1)^(1/2)/(-2*(a*(-4+a))^(1/2)-2*a+4)^(1/2))-(4+(a*(-4 +a))^(1/2)-a)/(2*(a*(-4+a))^(1/2)-2*a+4)^(1/2)*arctan(2*(x^5-1)^(1/2)/(2*( a*(-4+a))^(1/2)-2*a+4)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.43 \[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=\left [\frac {\log \left (\frac {x^{10} + 2 \, \sqrt {x^{5} - 1} \sqrt {a} x^{5} + a x^{5} - a}{x^{10} - a x^{5} + a}\right )}{5 \, \sqrt {a}}, -\frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {x^{5} - 1} \sqrt {-a} x^{5}}{a x^{5} - a}\right )}{5 \, a}\right ] \]
[1/5*log((x^10 + 2*sqrt(x^5 - 1)*sqrt(a)*x^5 + a*x^5 - a)/(x^10 - a*x^5 + a))/sqrt(a), -2/5*sqrt(-a)*arctan(sqrt(x^5 - 1)*sqrt(-a)*x^5/(a*x^5 - a))/ a]
Timed out. \[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=\int { -\frac {x^{9} - 2 \, x^{4}}{{\left (x^{10} - a x^{5} + a\right )} \sqrt {x^{5} - 1}} \,d x } \]
\[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=\int { -\frac {x^{9} - 2 \, x^{4}}{{\left (x^{10} - a x^{5} + a\right )} \sqrt {x^{5} - 1}} \,d x } \]
Time = 7.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.24 \[ \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx=\frac {\ln \left (\frac {a\,x^5-a+x^{10}+2\,\sqrt {a}\,x^5\,\sqrt {x^5-1}}{x^{10}-a\,x^5+a}\right )}{5\,\sqrt {a}} \]