Integrand size = 32, antiderivative size = 193 \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\frac {\sqrt {2} \left (1+\sqrt {1-4 a}-4 a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\sqrt {2} \left (-1+\sqrt {1-4 a}+4 a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{5 \sqrt {1-4 a} \sqrt {a} \sqrt {-1+\sqrt {1-4 a}+2 a}} \] Output:
1/5*2^(1/2)*(1+(1-4*a)^(1/2)-4*a)*arctan(2^(1/2)*a^(1/2)*(x^5-1)^(1/2)/(-1 -(1-4*a)^(1/2)+2*a)^(1/2))/(1-4*a)^(1/2)/a^(1/2)/(-1-(1-4*a)^(1/2)+2*a)^(1 /2)+1/5*2^(1/2)*(-1+(1-4*a)^(1/2)+4*a)*arctan(2^(1/2)*a^(1/2)*(x^5-1)^(1/2 )/(-1+(1-4*a)^(1/2)+2*a)^(1/2))/(1-4*a)^(1/2)/a^(1/2)/(-1+(1-4*a)^(1/2)+2* a)^(1/2)
Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\frac {\frac {\left (1+\sqrt {1-4 a}-4 a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1-\sqrt {1-4 a}+2 a}}\right )}{\sqrt {-1-\sqrt {1-4 a}+2 a}}+\frac {\left (-1+\sqrt {1-4 a}+4 a\right ) \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-1+x^5}}{\sqrt {-1+\sqrt {1-4 a}+2 a}}\right )}{\sqrt {-1+\sqrt {1-4 a}+2 a}}}{5 \sqrt {\frac {1}{2}-2 a} \sqrt {a}} \] Input:
Integrate[(x^4*(-2 + x^5))/(Sqrt[-1 + x^5]*(1 - x^5 + a*x^10)),x]
Output:
(((1 + Sqrt[1 - 4*a] - 4*a)*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[- 1 - Sqrt[1 - 4*a] + 2*a]])/Sqrt[-1 - Sqrt[1 - 4*a] + 2*a] + ((-1 + Sqrt[1 - 4*a] + 4*a)*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-1 + x^5])/Sqrt[-1 + Sqrt[1 - 4 *a] + 2*a]])/Sqrt[-1 + Sqrt[1 - 4*a] + 2*a])/(5*Sqrt[1/2 - 2*a]*Sqrt[a])
Time = 0.67 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.39, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {7266, 25, 1197, 1478, 25, 27, 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.
\(\displaystyle \int \frac {x^4 \left (x^5-2\right )}{\sqrt {x^5-1} \left (a x^{10}-x^5+1\right )} \, dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {1}{5} \int -\frac {2-x^5}{\sqrt {x^5-1} \left (a x^{10}-x^5+1\right )}dx^5\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{5} \int \frac {2-x^5}{\sqrt {x^5-1} \left (a x^{10}-x^5+1\right )}dx^5\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle -\frac {2}{5} \int \frac {1-x^{10}}{a x^{20}-(1-2 a) x^{10}+a}d\sqrt {x^5-1}\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle -\frac {2}{5} \left (-\frac {\int -\frac {1-2 \sqrt {a} \sqrt {x^5-1}}{\sqrt {a} \left (x^{10}-\frac {\sqrt {x^5-1}}{\sqrt {a}}+1\right )}d\sqrt {x^5-1}}{2 \sqrt {a}}-\frac {\int -\frac {2 \sqrt {a} \sqrt {x^5-1}+1}{\sqrt {a} \left (x^{10}+\frac {\sqrt {x^5-1}}{\sqrt {a}}+1\right )}d\sqrt {x^5-1}}{2 \sqrt {a}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2}{5} \left (\frac {\int \frac {1-2 \sqrt {a} \sqrt {x^5-1}}{\sqrt {a} \left (x^{10}-\frac {\sqrt {x^5-1}}{\sqrt {a}}+1\right )}d\sqrt {x^5-1}}{2 \sqrt {a}}+\frac {\int \frac {2 \sqrt {a} \sqrt {x^5-1}+1}{\sqrt {a} \left (x^{10}+\frac {\sqrt {x^5-1}}{\sqrt {a}}+1\right )}d\sqrt {x^5-1}}{2 \sqrt {a}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{5} \left (\frac {\int \frac {1-2 \sqrt {a} \sqrt {x^5-1}}{x^{10}-\frac {\sqrt {x^5-1}}{\sqrt {a}}+1}d\sqrt {x^5-1}}{2 a}+\frac {\int \frac {2 \sqrt {a} \sqrt {x^5-1}+1}{x^{10}+\frac {\sqrt {x^5-1}}{\sqrt {a}}+1}d\sqrt {x^5-1}}{2 a}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {2}{5} \left (\frac {\log \left (\sqrt {a} x^{10}+\sqrt {a}+\sqrt {x^5-1}\right )}{2 \sqrt {a}}-\frac {\log \left (\sqrt {a} x^{10}+\sqrt {a}-\sqrt {x^5-1}\right )}{2 \sqrt {a}}\right )\) |
Input:
Int[(x^4*(-2 + x^5))/(Sqrt[-1 + x^5]*(1 - x^5 + a*x^10)),x]
Output:
(-2*(-1/2*Log[Sqrt[a] + Sqrt[a]*x^10 - Sqrt[-1 + x^5]]/Sqrt[a] + Log[Sqrt[ a] + Sqrt[a]*x^10 + Sqrt[-1 + x^5]]/(2*Sqrt[a])))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(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 = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.22
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\sqrt {a}\, x^{5}-\sqrt {x^{5}-1}\right )-\ln \left (\sqrt {a}\, x^{5}+\sqrt {x^{5}-1}\right )}{5 \sqrt {a}}\) | \(43\) |
Input:
int(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x,method=_RETURNVERBOSE)
Output:
1/5*(ln(a^(1/2)*x^5-(x^5-1)^(1/2))-ln(a^(1/2)*x^5+(x^5-1)^(1/2)))/a^(1/2)
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.38 \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\left [\frac {\log \left (\frac {a x^{10} - 2 \, \sqrt {x^{5} - 1} \sqrt {a} x^{5} + x^{5} - 1}{a x^{10} - x^{5} + 1}\right )}{5 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{5}}{\sqrt {x^{5} - 1}}\right )}{5 \, a}\right ] \] Input:
integrate(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x, algorithm="fricas")
Output:
[1/5*log((a*x^10 - 2*sqrt(x^5 - 1)*sqrt(a)*x^5 + x^5 - 1)/(a*x^10 - x^5 + 1))/sqrt(a), 2/5*sqrt(-a)*arctan(sqrt(-a)*x^5/sqrt(x^5 - 1))/a]
Timed out. \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\text {Timed out} \] Input:
integrate(x**4*(x**5-2)/(x**5-1)**(1/2)/(a*x**10-x**5+1),x)
Output:
Timed out
\[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\int { \frac {{\left (x^{5} - 2\right )} x^{4}}{{\left (a x^{10} - x^{5} + 1\right )} \sqrt {x^{5} - 1}} \,d x } \] Input:
integrate(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x, algorithm="maxima")
Output:
integrate((x^5 - 2)*x^4/((a*x^10 - x^5 + 1)*sqrt(x^5 - 1)), x)
Time = 0.70 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.18 \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\frac {\pi + 2 \, \arctan \left (\frac {{\left (x^{5} - 1\right )} a + a}{\sqrt {x^{5} - 1} \sqrt {-a}}\right )}{5 \, \sqrt {-a}} \] Input:
integrate(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x, algorithm="giac")
Output:
1/5*(pi + 2*arctan(((x^5 - 1)*a + a)/(sqrt(x^5 - 1)*sqrt(-a))))/sqrt(-a)
Time = 10.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.24 \[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\frac {\ln \left (\frac {a\,x^{10}+x^5-2\,\sqrt {a}\,x^5\,\sqrt {x^5-1}-1}{4\,a\,x^{10}-4\,x^5+4}\right )}{5\,\sqrt {a}} \] Input:
int((x^4*(x^5 - 2))/((x^5 - 1)^(1/2)*(a*x^10 - x^5 + 1)),x)
Output:
log((a*x^10 + x^5 - 2*a^(1/2)*x^5*(x^5 - 1)^(1/2) - 1)/(4*a*x^10 - 4*x^5 + 4))/(5*a^(1/2))
\[ \int \frac {x^4 \left (-2+x^5\right )}{\sqrt {-1+x^5} \left (1-x^5+a x^{10}\right )} \, dx=\int \frac {\sqrt {x^{5}-1}\, x^{9}}{a \,x^{15}-a \,x^{10}-x^{10}+2 x^{5}-1}d x -2 \left (\int \frac {\sqrt {x^{5}-1}\, x^{4}}{a \,x^{15}-a \,x^{10}-x^{10}+2 x^{5}-1}d x \right ) \] Input:
int(x^4*(x^5-2)/(x^5-1)^(1/2)/(a*x^10-x^5+1),x)
Output:
int((sqrt(x**5 - 1)*x**9)/(a*x**15 - a*x**10 - x**10 + 2*x**5 - 1),x) - 2* int((sqrt(x**5 - 1)*x**4)/(a*x**15 - a*x**10 - x**10 + 2*x**5 - 1),x)