\(\int \frac {-1+4 x^5}{(1+x+x^5)^2} \, dx\) [139]

Optimal result

Integrand size = 16, antiderivative size = 11 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{1+x+x^5} \]

[Out]

-x/(x^5+x+1)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1602} \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{x^5+x+1} \]

[In]

Int[(-1 + 4*x^5)/(1 + x + x^5)^2,x]

[Out]

-(x/(1 + x + x^5))

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x}{1+x+x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{1+x+x^5} \]

[In]

Integrate[(-1 + 4*x^5)/(1 + x + x^5)^2,x]

[Out]

-(x/(1 + x + x^5))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {x}{x^{5}+x +1}\)	\(12\)
norman	\(-\frac {x}{x^{5}+x +1}\)	\(12\)
risch	\(-\frac {x}{x^{5}+x +1}\)	\(12\)
parallelrisch	\(-\frac {x}{x^{5}+x +1}\)	\(12\)
default	\(-\frac {-3 x^{2}+5 x -1}{7 \left (x^{3}-x^{2}+1\right )}+\frac {-3 x -1}{7 x^{2}+7 x +7}\)	\(41\)

[In]

int((4*x^5-1)/(x^5+x+1)^2,x,method=_RETURNVERBOSE)

[Out]

-x/(x^5+x+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{x^{5} + x + 1} \]

[In]

integrate((4*x^5-1)/(x^5+x+1)^2,x, algorithm="fricas")

[Out]

-x/(x^5 + x + 1)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=- \frac {x}{x^{5} + x + 1} \]

[In]

integrate((4*x**5-1)/(x**5+x+1)**2,x)

[Out]

-x/(x**5 + x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{x^{5} + x + 1} \]

[In]

integrate((4*x^5-1)/(x^5+x+1)^2,x, algorithm="maxima")

[Out]

-x/(x^5 + x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{x^{5} + x + 1} \]

[In]

integrate((4*x^5-1)/(x^5+x+1)^2,x, algorithm="giac")

[Out]

-x/(x^5 + x + 1)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx=-\frac {x}{x^5+x+1} \]

[In]

int((4*x^5 - 1)/(x + x^5 + 1)^2,x)

[Out]

-x/(x + x^5 + 1)