### 3.2169 $$\int \frac{(3-4 x+x^2)^2}{x^2} \, dx$$

Optimal. Leaf size=25 $\frac{x^3}{3}-4 x^2+22 x-\frac{9}{x}-24 \log (x)$

[Out]

-9/x + 22*x - 4*x^2 + x^3/3 - 24*Log[x]

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Rubi [A]  time = 0.01157, antiderivative size = 25, 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} $\frac{x^3}{3}-4 x^2+22 x-\frac{9}{x}-24 \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - 4*x + x^2)^2/x^2,x]

[Out]

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

Mathematica [A]  time = 0.00066, size = 25, normalized size = 1. $\frac{x^3}{3}-4 x^2+22 x-\frac{9}{x}-24 \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 - 4*x + x^2)^2/x^2,x]

[Out]

-9/x + 22*x - 4*x^2 + x^3/3 - 24*Log[x]

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Maple [A]  time = 0.044, size = 24, normalized size = 1. \begin{align*} -9\,{x}^{-1}+22\,x-4\,{x}^{2}+{\frac{{x}^{3}}{3}}-24\,\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^2,x)

[Out]

-9/x+22*x-4*x^2+1/3*x^3-24*ln(x)

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Maxima [A]  time = 0.954974, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{3} \, x^{3} - 4 \, x^{2} + 22 \, x - \frac{9}{x} - 24 \, \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^2,x, algorithm="maxima")

[Out]

1/3*x^3 - 4*x^2 + 22*x - 9/x - 24*log(x)

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Fricas [A]  time = 1.68577, size = 68, normalized size = 2.72 \begin{align*} \frac{x^{4} - 12 \, x^{3} + 66 \, x^{2} - 72 \, x \log \left (x\right ) - 27}{3 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^4 - 12*x^3 + 66*x^2 - 72*x*log(x) - 27)/x

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Sympy [A]  time = 0.087531, size = 20, normalized size = 0.8 \begin{align*} \frac{x^{3}}{3} - 4 x^{2} + 22 x - 24 \log{\left (x \right )} - \frac{9}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 - 4*x**2 + 22*x - 24*log(x) - 9/x

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Giac [A]  time = 1.10401, size = 32, normalized size = 1.28 \begin{align*} \frac{1}{3} \, x^{3} - 4 \, x^{2} + 22 \, x - \frac{9}{x} - 24 \, \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^2,x, algorithm="giac")

[Out]

1/3*x^3 - 4*x^2 + 22*x - 9/x - 24*log(abs(x))