3.2181 \(\int \frac{1+2 x+x^2}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{x^2}-\frac{1}{3 x^3}-\frac{1}{x} \]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

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Rubi [A]  time = 0.0047827, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {14} \[ -\frac{1}{x^2}-\frac{1}{3 x^3}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x + x^2)/x^4,x]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.000654, size = 18, normalized size = 1. \[ -\frac{1}{x^2}-\frac{1}{3 x^3}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + x^2)/x^4,x]

[Out]

-1/(3*x^3) - x^(-2) - x^(-1)

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Maple [A]  time = 0.042, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{x}^{-2}-{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/x^3-1/x^2-1/x

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Maxima [A]  time = 1.00815, size = 20, normalized size = 1.11 \begin{align*} -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3

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Fricas [A]  time = 1.67771, size = 38, normalized size = 2.11 \begin{align*} -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3

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Sympy [A]  time = 0.087984, size = 15, normalized size = 0.83 \begin{align*} - \frac{3 x^{2} + 3 x + 1}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x**2 + 3*x + 1)/(3*x**3)

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Giac [A]  time = 1.13123, size = 20, normalized size = 1.11 \begin{align*} -\frac{3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*x^2 + 3*x + 1)/x^3