3.2173 $$\int \frac{(3-4 x+x^2)^2}{x^6} \, dx$$

Optimal. Leaf size=30 $\frac{4}{x^2}-\frac{22}{3 x^3}+\frac{6}{x^4}-\frac{9}{5 x^5}-\frac{1}{x}$

[Out]

-9/(5*x^5) + 6/x^4 - 22/(3*x^3) + 4/x^2 - x^(-1)

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Rubi [A]  time = 0.0116723, antiderivative size = 30, 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} $\frac{4}{x^2}-\frac{22}{3 x^3}+\frac{6}{x^4}-\frac{9}{5 x^5}-\frac{1}{x}$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - 4*x + x^2)^2/x^6,x]

[Out]

-9/(5*x^5) + 6/x^4 - 22/(3*x^3) + 4/x^2 - x^(-1)

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^6} \, dx &=\int \left (\frac{9}{x^6}-\frac{24}{x^5}+\frac{22}{x^4}-\frac{8}{x^3}+\frac{1}{x^2}\right ) \, dx\\ &=-\frac{9}{5 x^5}+\frac{6}{x^4}-\frac{22}{3 x^3}+\frac{4}{x^2}-\frac{1}{x}\\ \end{align*}

Mathematica [A]  time = 0.0006086, size = 30, normalized size = 1. $\frac{4}{x^2}-\frac{22}{3 x^3}+\frac{6}{x^4}-\frac{9}{5 x^5}-\frac{1}{x}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 - 4*x + x^2)^2/x^6,x]

[Out]

-9/(5*x^5) + 6/x^4 - 22/(3*x^3) + 4/x^2 - x^(-1)

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Maple [A]  time = 0.043, size = 27, normalized size = 0.9 \begin{align*} -{\frac{9}{5\,{x}^{5}}}+6\,{x}^{-4}-{\frac{22}{3\,{x}^{3}}}+4\,{x}^{-2}-{x}^{-1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-4*x+3)^2/x^6,x)

[Out]

-9/5/x^5+6/x^4-22/3/x^3+4/x^2-1/x

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Maxima [A]  time = 1.00619, size = 34, normalized size = 1.13 \begin{align*} -\frac{15 \, x^{4} - 60 \, x^{3} + 110 \, x^{2} - 90 \, x + 27}{15 \, x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^6,x, algorithm="maxima")

[Out]

-1/15*(15*x^4 - 60*x^3 + 110*x^2 - 90*x + 27)/x^5

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Fricas [A]  time = 1.88511, size = 69, normalized size = 2.3 \begin{align*} -\frac{15 \, x^{4} - 60 \, x^{3} + 110 \, x^{2} - 90 \, x + 27}{15 \, x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*x^4 - 60*x^3 + 110*x^2 - 90*x + 27)/x^5

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Sympy [A]  time = 0.105595, size = 26, normalized size = 0.87 \begin{align*} - \frac{15 x^{4} - 60 x^{3} + 110 x^{2} - 90 x + 27}{15 x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-4*x+3)**2/x**6,x)

[Out]

-(15*x**4 - 60*x**3 + 110*x**2 - 90*x + 27)/(15*x**5)

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Giac [A]  time = 1.11944, size = 34, normalized size = 1.13 \begin{align*} -\frac{15 \, x^{4} - 60 \, x^{3} + 110 \, x^{2} - 90 \, x + 27}{15 \, x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^6,x, algorithm="giac")

[Out]

-1/15*(15*x^4 - 60*x^3 + 110*x^2 - 90*x + 27)/x^5