3.19.41 \(\int \frac {(3-4 x+x^2)^2}{x^5} \, dx\)

Optimal. Leaf size=25 \[ -\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {698} \begin {gather*} -\frac {11}{x^2}+\frac {8}{x^3}-\frac {9}{4 x^4}+\frac {8}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x + x^2)^2/x^5,x]

[Out]

-9/(4*x^4) + 8/x^3 - 11/x^2 + 8/x + Log[x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx &=\int \left (\frac {9}{x^5}-\frac {24}{x^4}+\frac {22}{x^3}-\frac {8}{x^2}+\frac {1}{x}\right ) \, dx\\ &=-\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x + x^2)^2/x^5,x]

[Out]

-9/(4*x^4) + 8/x^3 - 11/x^2 + 8/x + Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 - 4*x + x^2)^2/x^5,x]

[Out]

IntegrateAlgebraic[(3 - 4*x + x^2)^2/x^5, x]

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fricas [A]  time = 0.38, size = 27, normalized size = 1.08 \begin {gather*} \frac {4 \, x^{4} \log \relax (x) + 32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(4*x^4*log(x) + 32*x^3 - 44*x^2 + 32*x - 9)/x^4

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giac [A]  time = 0.15, size = 24, normalized size = 0.96 \begin {gather*} \frac {32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^5,x, algorithm="giac")

[Out]

1/4*(32*x^3 - 44*x^2 + 32*x - 9)/x^4 + log(abs(x))

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maple [A]  time = 0.05, size = 24, normalized size = 0.96 \begin {gather*} \ln \relax (x )+\frac {8}{x}-\frac {11}{x^{2}}+\frac {8}{x^{3}}-\frac {9}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-4*x+3)^2/x^5,x)

[Out]

-9/4/x^4+8/x^3-11/x^2+8/x+ln(x)

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maxima [A]  time = 0.99, size = 23, normalized size = 0.92 \begin {gather*} \frac {32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^5,x, algorithm="maxima")

[Out]

1/4*(32*x^3 - 44*x^2 + 32*x - 9)/x^4 + log(x)

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mupad [B]  time = 0.03, size = 22, normalized size = 0.88 \begin {gather*} \ln \relax (x)+\frac {8\,x^3-11\,x^2+8\,x-\frac {9}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 4*x + 3)^2/x^5,x)

[Out]

log(x) + (8*x - 11*x^2 + 8*x^3 - 9/4)/x^4

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sympy [A]  time = 0.11, size = 22, normalized size = 0.88 \begin {gather*} \log {\relax (x )} + \frac {32 x^{3} - 44 x^{2} + 32 x - 9}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-4*x+3)**2/x**5,x)

[Out]

log(x) + (32*x**3 - 44*x**2 + 32*x - 9)/(4*x**4)

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