3.2171 $$\int \frac{(3-4 x+x^2)^2}{x^4} \, dx$$

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

[Out]

-3/x^3 + 12/x^2 - 22/x + x - 8*Log[x]

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Rubi [A]  time = 0.0107885, antiderivative size = 21, 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} $\frac{12}{x^2}-\frac{3}{x^3}+x-\frac{22}{x}-8 \log (x)$

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 = 0.0006489, size = 21, normalized size = 1. $\frac{12}{x^2}-\frac{3}{x^3}+x-\frac{22}{x}-8 \log (x)$

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]

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Maple [A]  time = 0.045, size = 22, normalized size = 1.1 \begin{align*} -3\,{x}^{-3}+12\,{x}^{-2}-22\,{x}^{-1}+x-8\,\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^4,x)

[Out]

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

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Maxima [A]  time = 0.985285, size = 28, normalized size = 1.33 \begin{align*} x - \frac{22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6889, size = 62, normalized size = 2.95 \begin{align*} \frac{x^{4} - 8 \, x^{3} \log \left (x\right ) - 22 \, x^{2} + 12 \, x - 3}{x^{3}} \end{align*}

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

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Sympy [A]  time = 0.100731, size = 19, normalized size = 0.9 \begin{align*} x - 8 \log{\left (x \right )} - \frac{22 x^{2} - 12 x + 3}{x^{3}} \end{align*}

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

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Giac [A]  time = 1.13196, size = 30, normalized size = 1.43 \begin{align*} x - \frac{22 \, x^{2} - 12 \, x + 3}{x^{3}} - 8 \, \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^4,x, algorithm="giac")

[Out]

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