### 3.2168 $$\int \frac{(3-4 x+x^2)^2}{x} \, dx$$

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

[Out]

-24*x + 11*x^2 - (8*x^3)/3 + x^4/4 + 9*Log[x]

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Rubi [A]  time = 0.0092649, antiderivative size = 27, normalized size of antiderivative = 1., 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} $\frac{x^4}{4}-\frac{8 x^3}{3}+11 x^2-24 x+9 \log (x)$

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*}

Mathematica [A]  time = 0.0006272, size = 27, normalized size = 1. $\frac{x^4}{4}-\frac{8 x^3}{3}+11 x^2-24 x+9 \log (x)$

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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Maple [A]  time = 0.038, size = 24, normalized size = 0.9 \begin{align*} -24\,x+11\,{x}^{2}-{\frac{8\,{x}^{3}}{3}}+{\frac{{x}^{4}}{4}}+9\,\ln \left ( x \right ) \end{align*}

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 = 0.960832, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \frac{8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.63101, size = 62, normalized size = 2.3 \begin{align*} \frac{1}{4} \, x^{4} - \frac{8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.082447, size = 24, normalized size = 0.89 \begin{align*} \frac{x^{4}}{4} - \frac{8 x^{3}}{3} + 11 x^{2} - 24 x + 9 \log{\left (x \right )} \end{align*}

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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Giac [A]  time = 1.14016, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{4} \, x^{4} - \frac{8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left ({\left | x \right |}\right ) \end{align*}

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))