\(\int \frac {(3-4 x+x^2)^2}{x} \, dx\) [2168]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 14, antiderivative size = 27 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=-24 x+11 x^2-\frac {8 x^3}{3}+\frac {x^4}{4}+9 \log (x) \]

[Out]

-24*x+11*x^2-8/3*x^3+1/4*x^4+9*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} \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=\frac {x^4}{4}-\frac {8 x^3}{3}+11 x^2-24 x+9 \log (x) \]

[In]

Int[(3 - 4*x + x^2)^2/x,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=-24 x+11 x^2-\frac {8 x^3}{3}+\frac {x^4}{4}+9 \log (x) \]

[In]

Integrate[(3 - 4*x + x^2)^2/x,x]

[Out]

-24*x + 11*x^2 - (8*x^3)/3 + x^4/4 + 9*Log[x]

Maple [A] (verified)

Time = 16.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

-24*x+11*x^2-8/3*x^3+1/4*x^4+9*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=\frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left (x\right ) \]

[In]

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

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=\frac {x^{4}}{4} - \frac {8 x^{3}}{3} + 11 x^{2} - 24 x + 9 \log {\left (x \right )} \]

[In]

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

[Out]

x**4/4 - 8*x**3/3 + 11*x**2 - 24*x + 9*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=\frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left (x\right ) \]

[In]

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

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=\frac {1}{4} \, x^{4} - \frac {8}{3} \, x^{3} + 11 \, x^{2} - 24 \, x + 9 \, \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

1/4*x^4 - 8/3*x^3 + 11*x^2 - 24*x + 9*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (3-4 x+x^2\right )^2}{x} \, dx=9\,\ln \left (x\right )-24\,x+11\,x^2-\frac {8\,x^3}{3}+\frac {x^4}{4} \]

[In]

int((x^2 - 4*x + 3)^2/x,x)

[Out]

9*log(x) - 24*x + 11*x^2 - (8*x^3)/3 + x^4/4