### 3.2170 $$\int \frac{(3-4 x+x^2)^2}{x^3} \, dx$$

Optimal. Leaf size=27 $\frac{x^2}{2}-\frac{9}{2 x^2}-8 x+\frac{24}{x}+22 \log (x)$

[Out]

-9/(2*x^2) + 24/x - 8*x + x^2/2 + 22*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0113032, 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} $\frac{x^2}{2}-\frac{9}{2 x^2}-8 x+\frac{24}{x}+22 \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 - 4*x + x^2)^2/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0007447, size = 27, normalized size = 1. $\frac{x^2}{2}-\frac{9}{2 x^2}-8 x+\frac{24}{x}+22 \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 - 4*x + x^2)^2/x^3,x]

[Out]

-9/(2*x^2) + 24/x - 8*x + x^2/2 + 22*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 24, normalized size = 0.9 \begin{align*} -{\frac{9}{2\,{x}^{2}}}+24\,{x}^{-1}-8\,x+{\frac{{x}^{2}}{2}}+22\,\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^3,x)

[Out]

-9/2/x^2+24/x-8*x+1/2*x^2+22*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 1.06307, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} - 8 \, x + \frac{3 \,{\left (16 \, x - 3\right )}}{2 \, x^{2}} + 22 \, \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^3,x, algorithm="maxima")

[Out]

1/2*x^2 - 8*x + 3/2*(16*x - 3)/x^2 + 22*log(x)

________________________________________________________________________________________

Fricas [A]  time = 1.72387, size = 69, normalized size = 2.56 \begin{align*} \frac{x^{4} - 16 \, x^{3} + 44 \, x^{2} \log \left (x\right ) + 48 \, x - 9}{2 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^4 - 16*x^3 + 44*x^2*log(x) + 48*x - 9)/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.09643, size = 22, normalized size = 0.81 \begin{align*} \frac{x^{2}}{2} - 8 x + 22 \log{\left (x \right )} + \frac{48 x - 9}{2 x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - 8*x + 22*log(x) + (48*x - 9)/(2*x**2)

________________________________________________________________________________________

Giac [A]  time = 1.14068, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, x^{2} - 8 \, x + \frac{3 \,{\left (16 \, x - 3\right )}}{2 \, x^{2}} + 22 \, \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^3,x, algorithm="giac")

[Out]

1/2*x^2 - 8*x + 3/2*(16*x - 3)/x^2 + 22*log(abs(x))