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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {712} \begin {gather*} \frac {x^2}{2}-\frac {9}{2 x^2}-8 x+\frac {24}{x}+22 \log (x) \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {9}{2 x^2}+\frac {24}{x}-8 x+\frac {x^2}{2}+22 \log (x) \end {gather*}

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]

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Maple [A]
time = 0.72, size = 24, normalized size = 0.89

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	\(\frac {-\frac {9}{2}+24 x -8 x^{3}+\frac {1}{2} x^{4}}{x^{2}}+22 \ln \left (x \right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, x^{2} - 8 \, x + \frac {3 \, {\left (16 \, x - 3\right )}}{2 \, x^{2}} + 22 \, \log \left (x\right ) \end {gather*}

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)

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Fricas [A]
time = 3.15, size = 25, normalized size = 0.93 \begin {gather*} \frac {x^{4} - 16 \, x^{3} + 44 \, x^{2} \log \left (x\right ) + 48 \, x - 9}{2 \, x^{2}} \end {gather*}

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

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Sympy [A]
time = 0.03, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{2} - 8 x + 22 \log {\left (x \right )} + \frac {48 x - 9}{2 x^{2}} \end {gather*}

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)

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Giac [A]
time = 0.97, size = 24, normalized size = 0.89 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.03, size = 22, normalized size = 0.81 \begin {gather*} 22\,\ln \left (x\right )-8\,x+\frac {24\,x-\frac {9}{2}}{x^2}+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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