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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{x^2}-\frac {1}{3 x^3}-\frac {1}{x} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{3 x^3}-\frac {1}{x^2}-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + x^2)/x^4,x]

[Out]

-1/3*1/x^3 - x^(-2) - x^(-1)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x+x^2}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + 2*x + x^2)/x^4,x]

[Out]

IntegrateAlgebraic[(1 + 2*x + x^2)/x^4, x]

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fricas [A]  time = 0.38, size = 15, normalized size = 0.83 \begin {gather*} -\frac {3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end {gather*}

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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giac [A]  time = 0.15, size = 15, normalized size = 0.83 \begin {gather*} -\frac {3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end {gather*}

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

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maple [A]  time = 0.05, size = 17, normalized size = 0.94 \begin {gather*} -\frac {1}{x}-\frac {1}{x^{2}}-\frac {1}{3 x^{3}} \end {gather*}

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 = 0.87, size = 15, normalized size = 0.83 \begin {gather*} -\frac {3 \, x^{2} + 3 \, x + 1}{3 \, x^{3}} \end {gather*}

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.61 \begin {gather*} -\frac {x^2+x+\frac {1}{3}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2 + 1)/x^4,x)

[Out]

-(x + x^2 + 1/3)/x^3

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sympy [A]  time = 0.10, size = 15, normalized size = 0.83 \begin {gather*} \frac {- 3 x^{2} - 3 x - 1}{3 x^{3}} \end {gather*}

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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