\(\int \frac {(3-4 x+x^2)^2}{x^5} \, dx\) [2172]

Optimal result

Integrand size = 14, antiderivative size = 25 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=-\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {712} \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=-\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x) \]

[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 712

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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=-\frac {9}{4 x^4}+\frac {8}{x^3}-\frac {11}{x^2}+\frac {8}{x}+\log (x) \]

[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]

Maple [A] (verified)

Time = 13.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

method	result	size
norman	\(\frac {-\frac {9}{4}+8 x^{3}-11 x^{2}+8 x}{x^{4}}+\ln \left (x \right )\)	\(23\)
risch	\(\frac {-\frac {9}{4}+8 x^{3}-11 x^{2}+8 x}{x^{4}}+\ln \left (x \right )\)	\(23\)
default	\(-\frac {9}{4 x^{4}}+\frac {8}{x^{3}}-\frac {11}{x^{2}}+\frac {8}{x}+\ln \left (x \right )\)	\(24\)
parallelrisch	\(\frac {4 \ln \left (x \right ) x^{4}-9+32 x^{3}-44 x^{2}+32 x}{4 x^{4}}\)	\(28\)

[In]

int((x^2-4*x+3)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=\frac {4 \, x^{4} \log \left (x\right ) + 32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=\log {\left (x \right )} + \frac {32 x^{3} - 44 x^{2} + 32 x - 9}{4 x^{4}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=\frac {32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \log \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=\frac {32 \, x^{3} - 44 \, x^{2} + 32 \, x - 9}{4 \, x^{4}} + \log \left ({\left | x \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^5} \, dx=\ln \left (x\right )+\frac {8\,x^3-11\,x^2+8\,x-\frac {9}{4}}{x^4} \]

[In]

int((x^2 - 4*x + 3)^2/x^5,x)

[Out]

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