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/(x^5+x+1)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = 0.01, size = 11, normalized size = 1.00 \[ -\frac {x}{x^5+x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/(1 + x + x^5))

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 11, normalized size = 1.00 \[ -\frac {x}{x^{5} + x + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.33, size = 11, normalized size = 1.00 \[ -\frac {x}{x^{5} + x + 1} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 41, normalized size = 3.73 \[ -\frac {-3 x^{2}+5 x -1}{7 \left (x^{3}-x^{2}+1\right )}+\frac {-3 x -1}{7 x^{2}+7 x +7} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 11, normalized size = 1.00 \[ -\frac {x}{x^{5} + x + 1} \]

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)

________________________________________________________________________________________

mupad [B]  time = 2.30, size = 11, normalized size = 1.00 \[ -\frac {x}{x^5+x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5 - 1)/(x + x^5 + 1)^2,x)

[Out]

-x/(x + x^5 + 1)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 8, normalized size = 0.73 \[ - \frac {x}{x^{5} + x + 1} \]

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)

________________________________________________________________________________________