\(\int \frac {(3-4 x+x^2)^2}{x^3} \, dx\) [2170]

Optimal result

Integrand size = 14, antiderivative size = 27 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=-\frac {9}{2 x^2}+\frac {24}{x}-8 x+\frac {x^2}{2}+22 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, 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} \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=\frac {x^2}{2}-\frac {9}{2 x^2}-8 x+\frac {24}{x}+22 \log (x) \]

[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 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 (-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] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=-\frac {9}{2 x^2}+\frac {24}{x}-8 x+\frac {x^2}{2}+22 \log (x) \]

[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] (verified)

Time = 17.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {x^{2}}{2}-8 x +\frac {24 x -\frac {9}{2}}{x^{2}}+22 \ln \left (x \right )\)	\(23\)
default	\(-\frac {9}{2 x^{2}}+\frac {24}{x}-8 x +\frac {x^{2}}{2}+22 \ln \left (x \right )\)	\(24\)
norman	\(\frac {-\frac {9}{2}+24 x -8 x^{3}+\frac {1}{2} x^{4}}{x^{2}}+22 \ln \left (x \right )\)	\(25\)
parallelrisch	\(\frac {x^{4}+44 \ln \left (x \right ) x^{2}-16 x^{3}-9+48 x}{2 x^{2}}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=\frac {x^{4} - 16 \, x^{3} + 44 \, x^{2} \log \left (x\right ) + 48 \, x - 9}{2 \, x^{2}} \]

[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] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=\frac {x^{2}}{2} - 8 x + 22 \log {\left (x \right )} + \frac {48 x - 9}{2 x^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=\frac {1}{2} \, x^{2} - 8 \, x + \frac {3 \, {\left (16 \, x - 3\right )}}{2 \, x^{2}} + 22 \, \log \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=\frac {1}{2} \, x^{2} - 8 \, x + \frac {3 \, {\left (16 \, x - 3\right )}}{2 \, x^{2}} + 22 \, \log \left ({\left | x \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x^3} \, dx=22\,\ln \left (x\right )-8\,x+\frac {24\,x-\frac {9}{2}}{x^2}+\frac {x^2}{2} \]

[In]

int((x^2 - 4*x + 3)^2/x^3,x)

[Out]

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