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

Optimal. Leaf size=21 \[ -\frac {3}{x^3}+\frac {12}{x^2}+x-\frac {22}{x}-8 \log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, 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 {12}{x^2}-\frac {3}{x^3}+x-\frac {22}{x}-8 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x + x^2)^2/x^4,x]

[Out]

-3/x^3 + 12/x^2 - 22/x + x - 8*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^4} \, dx &=\int \left (1+\frac {9}{x^4}-\frac {24}{x^3}+\frac {22}{x^2}-\frac {8}{x}\right ) \, dx\\ &=-\frac {3}{x^3}+\frac {12}{x^2}-\frac {22}{x}+x-8 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {3}{x^3}+\frac {12}{x^2}+x-\frac {22}{x}-8 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x + x^2)^2/x^4,x]

[Out]

-3/x^3 + 12/x^2 - 22/x + x - 8*Log[x]

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 24, normalized size = 1.14 \begin {gather*} \frac {x^{4} - 8 \, x^{3} \log \relax (x) - 22 \, x^{2} + 12 \, x - 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4 - 8*x^3*log(x) - 22*x^2 + 12*x - 3)/x^3

________________________________________________________________________________________

giac [A]  time = 0.16, size = 22, normalized size = 1.05 \begin {gather*} x - \frac {22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \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^4,x, algorithm="giac")

[Out]

x - (22*x^2 - 12*x + 3)/x^3 - 8*log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 22, normalized size = 1.05 \begin {gather*} x -8 \ln \relax (x )-\frac {22}{x}+\frac {12}{x^{2}}-\frac {3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/x^3+12/x^2-22/x+x-8*ln(x)

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 21, normalized size = 1.00 \begin {gather*} x - \frac {22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - (22*x^2 - 12*x + 3)/x^3 - 8*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 21, normalized size = 1.00 \begin {gather*} x-8\,\ln \relax (x)-\frac {22\,x^2-12\,x+3}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 4*x + 3)^2/x^4,x)

[Out]

x - 8*log(x) - (22*x^2 - 12*x + 3)/x^3

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 19, normalized size = 0.90 \begin {gather*} x - 8 \log {\relax (x )} + \frac {- 22 x^{2} + 12 x - 3}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 8*log(x) + (-22*x**2 + 12*x - 3)/x**3

________________________________________________________________________________________