3.118 \(\int \frac{1+2 x^2}{x^5 (1+x^2)^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Rubi [A]  time = 0.0083808, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {446, 74} \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(x^5*(1 + x^2)^3),x]

[Out]

-1/(4*x^4*(1 + x^2)^2)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int \frac{1+2 x^2}{x^5 \left (1+x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+2 x}{x^3 (1+x)^3} \, dx,x,x^2\right )\\ &=-\frac{1}{4 x^4 \left (1+x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0058138, size = 14, normalized size = 1. \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2)/(x^5*(1 + x^2)^3),x]

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Maple [B]  time = 0.011, size = 30, normalized size = 2.1 \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^2+1)/x^5/(x^2+1)^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 = 0.981792, size = 22, normalized size = 1.57 \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^2+1)/x^5/(x^2+1)^3,x, algorithm="maxima")

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Fricas [A]  time = 1.22072, size = 35, normalized size = 2.5 \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^2+1)/x^5/(x^2+1)^3,x, algorithm="fricas")

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Sympy [A]  time = 0.127676, size = 17, normalized size = 1.21 \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**2+1)/x**5/(x**2+1)**3,x)

[Out]

-1/(4*x**8 + 8*x**6 + 4*x**4)

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Giac [A]  time = 1.14359, size = 15, normalized size = 1.07 \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^2+1)/x^5/(x^2+1)^3,x, algorithm="giac")

[Out]

-1/4/(x^4 + x^2)^2