### 3.2172 $$\int \frac{(3-4 x+x^2)^2}{x^5} \, dx$$

Optimal. Leaf size=25 $-\frac{11}{x^2}+\frac{8}{x^3}-\frac{9}{4 x^4}+\frac{8}{x}+\log (x)$

[Out]

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

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Rubi [A]  time = 0.0112481, 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} $-\frac{11}{x^2}+\frac{8}{x^3}-\frac{9}{4 x^4}+\frac{8}{x}+\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - 4*x + x^2)^2/x^5,x]

[Out]

-9/(4*x^4) + 8/x^3 - 11/x^2 + 8/x + 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^5} \, dx &=\int \left (\frac{9}{x^5}-\frac{24}{x^4}+\frac{22}{x^3}-\frac{8}{x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{9}{4 x^4}+\frac{8}{x^3}-\frac{11}{x^2}+\frac{8}{x}+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0006945, size = 25, normalized size = 1. $-\frac{11}{x^2}+\frac{8}{x^3}-\frac{9}{4 x^4}+\frac{8}{x}+\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 - 4*x + x^2)^2/x^5,x]

[Out]

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

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Maple [A]  time = 0.043, size = 24, normalized size = 1. \begin{align*} -{\frac{9}{4\,{x}^{4}}}+8\,{x}^{-3}-11\,{x}^{-2}+8\,{x}^{-1}+\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^5,x)

[Out]

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

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Maxima [A]  time = 0.995359, size = 31, normalized size = 1.24 \begin{align*} \frac{32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \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^5,x, algorithm="maxima")

[Out]

1/4*(32*x^3 - 44*x^2 + 32*x - 9)/x^4 + log(x)

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Fricas [A]  time = 1.85166, size = 72, normalized size = 2.88 \begin{align*} \frac{4 \, x^{4} \log \left (x\right ) + 32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(4*x^4*log(x) + 32*x^3 - 44*x^2 + 32*x - 9)/x^4

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Sympy [A]  time = 0.105897, size = 22, normalized size = 0.88 \begin{align*} \log{\left (x \right )} + \frac{32 x^{3} - 44 x^{2} + 32 x - 9}{4 x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-4*x+3)**2/x**5,x)

[Out]

log(x) + (32*x**3 - 44*x**2 + 32*x - 9)/(4*x**4)

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Giac [A]  time = 1.13567, size = 32, normalized size = 1.28 \begin{align*} \frac{32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \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^5,x, algorithm="giac")

[Out]

1/4*(32*x^3 - 44*x^2 + 32*x - 9)/x^4 + log(abs(x))