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

Optimal. Leaf size=27 \[ \frac {x^4}{4}-\frac {8 x^3}{3}+11 x^2-24 x+9 \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 27, 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 {x^4}{4}-\frac {8 x^3}{3}+11 x^2-24 x+9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x + x^2)^2/x,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^4}{4}-\frac {8 x^3}{3}+11 x^2-24 x+9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x + x^2)^2/x,x]

[Out]

-24*x + 11*x^2 - (8*x^3)/3 + x^4/4 + 9*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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(x)

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giac [A]  time = 0.15, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \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,x, algorithm="giac")

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(abs(x))

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maple [A]  time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{4}}{4}-\frac {8 x^{3}}{3}+11 x^{2}-24 x +9 \ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*x+11*x^2-8/3*x^3+1/4*x^4+9*ln(x)

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maxima [A]  time = 1.00, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(x)

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mupad [B]  time = 0.02, size = 23, normalized size = 0.85 \begin {gather*} 9\,\ln \relax (x)-24\,x+11\,x^2-\frac {8\,x^3}{3}+\frac {x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 4*x + 3)^2/x,x)

[Out]

9*log(x) - 24*x + 11*x^2 - (8*x^3)/3 + x^4/4

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sympy [A]  time = 0.09, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{4}}{4} - \frac {8 x^{3}}{3} + 11 x^{2} - 24 x + 9 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/4 - 8*x**3/3 + 11*x**2 - 24*x + 9*log(x)

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