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

Optimal. Leaf size=11 \[ -\frac{x}{x^5+x+1} \]

[Out]

-(x/(1 + x + x^5))

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Rubi [A]  time = 0.0068737, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1588} \[ -\frac{x}{x^5+x+1} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x^5)/(1 + x + x^5)^2,x]

[Out]

-(x/(1 + x + x^5))

Rule 1588

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*} \int \frac{-1+4 x^5}{\left (1+x+x^5\right )^2} \, dx &=-\frac{x}{1+x+x^5}\\ \end{align*}

Mathematica [A]  time = 0.0065788, size = 11, normalized size = 1. \[ -\frac{x}{x^5+x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/(1 + x + x^5))

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Maple [B]  time = 0.009, size = 41, normalized size = 3.7 \begin{align*} -{\frac{-3\,{x}^{2}+5\,x-1}{7\,{x}^{3}-7\,{x}^{2}+7}}+{\frac{-3\,x-1}{7\,{x}^{2}+7\,x+7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(-3*x^2+5*x-1)/(x^3-x^2+1)+1/7*(-3*x-1)/(x^2+x+1)

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Maxima [A]  time = 1.02207, size = 15, normalized size = 1.36 \begin{align*} -\frac{x}{x^{5} + x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^5 + x + 1)

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Fricas [A]  time = 1.35098, size = 24, normalized size = 2.18 \begin{align*} -\frac{x}{x^{5} + x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^5 + x + 1)

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Sympy [A]  time = 0.119274, size = 8, normalized size = 0.73 \begin{align*} - \frac{x}{x^{5} + x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x**5 + x + 1)

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Giac [A]  time = 1.24403, size = 15, normalized size = 1.36 \begin{align*} -\frac{x}{x^{5} + x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^5 + x + 1)