3.443 \(\int \frac{x+2 x^3}{(x^2+x^4)^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{4 \left (x^4+x^2\right )^2} \]

[Out]

-1/(4*(x^2 + x^4)^2)

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Rubi [A]  time = 0.0070267, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {1588} \[ -\frac{1}{4 \left (x^4+x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x + 2*x^3)/(x^2 + x^4)^3,x]

[Out]

-1/(4*(x^2 + x^4)^2)

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{x+2 x^3}{\left (x^2+x^4\right )^3} \, dx &=-\frac{1}{4 \left (x^2+x^4\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0065667, size = 14, normalized size = 1.08 \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x + 2*x^3)/(x^2 + x^4)^3,x]

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Maple [B]  time = 0.01, size = 30, normalized size = 2.3 \begin{align*} -{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}}-{\frac{1}{2\,{x}^{2}+2}}-{\frac{1}{4\,{x}^{4}}}+{\frac{1}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+x)/(x^4+x^2)^3,x)

[Out]

-1/4/(x^2+1)^2-1/2/(x^2+1)-1/4/x^4+1/2/x^2

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Maxima [A]  time = 0.967609, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{4 \,{\left (x^{4} + x^{2}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x)/(x^4+x^2)^3,x, algorithm="maxima")

[Out]

-1/4/(x^4 + x^2)^2

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Fricas [A]  time = 0.937095, size = 35, normalized size = 2.69 \begin{align*} -\frac{1}{4 \,{\left (x^{8} + 2 \, x^{6} + x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x)/(x^4+x^2)^3,x, algorithm="fricas")

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Sympy [A]  time = 0.130826, size = 17, normalized size = 1.31 \begin{align*} - \frac{1}{4 x^{8} + 8 x^{6} + 4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+x)/(x**4+x**2)**3,x)

[Out]

-1/(4*x**8 + 8*x**6 + 4*x**4)

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Giac [A]  time = 1.20607, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{4 \,{\left (x^{4} + x^{2}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x)/(x^4+x^2)^3,x, algorithm="giac")

[Out]

-1/4/(x^4 + x^2)^2