3.96.30 \(\int \frac {-1+44 x-40 x^2-6 x^3+2 x^4}{x} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{2} \left (-22-2 x+x^2\right )^2-\log (x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {14} \begin {gather*} \frac {x^4}{2}-2 x^3-20 x^2+44 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 44*x - 40*x^2 - 6*x^3 + 2*x^4)/x,x]

[Out]

44*x - 20*x^2 - 2*x^3 + x^4/2 - Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (44-\frac {1}{x}-40 x-6 x^2+2 x^3\right ) \, dx\\ &=44 x-20 x^2-2 x^3+\frac {x^4}{2}-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 25, normalized size = 1.32 \begin {gather*} 44 x-20 x^2-2 x^3+\frac {x^4}{2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 44*x - 40*x^2 - 6*x^3 + 2*x^4)/x,x]

[Out]

44*x - 20*x^2 - 2*x^3 + x^4/2 - Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, x^{4} - 2 \, x^{3} - 20 \, x^{2} + 44 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-6*x^3-40*x^2+44*x-1)/x,x, algorithm="fricas")

[Out]

1/2*x^4 - 2*x^3 - 20*x^2 + 44*x - log(x)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 24, normalized size = 1.26 \begin {gather*} \frac {1}{2} \, x^{4} - 2 \, x^{3} - 20 \, x^{2} + 44 \, x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-6*x^3-40*x^2+44*x-1)/x,x, algorithm="giac")

[Out]

1/2*x^4 - 2*x^3 - 20*x^2 + 44*x - log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 24, normalized size = 1.26

method	result	size
default	\(\frac {x^{4}}{2}-2 x^{3}-20 x^{2}+44 x -\ln \relax (x )\)	\(24\)
norman	\(\frac {x^{4}}{2}-2 x^{3}-20 x^{2}+44 x -\ln \relax (x )\)	\(24\)
risch	\(\frac {x^{4}}{2}-2 x^{3}-20 x^{2}+44 x -\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4-6*x^3-40*x^2+44*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

1/2*x^4-2*x^3-20*x^2+44*x-ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, x^{4} - 2 \, x^{3} - 20 \, x^{2} + 44 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4-6*x^3-40*x^2+44*x-1)/x,x, algorithm="maxima")

[Out]

1/2*x^4 - 2*x^3 - 20*x^2 + 44*x - log(x)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 23, normalized size = 1.21 \begin {gather*} 44\,x-\ln \relax (x)-20\,x^2-2\,x^3+\frac {x^4}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x^2 - 44*x + 6*x^3 - 2*x^4 + 1)/x,x)

[Out]

44*x - log(x) - 20*x^2 - 2*x^3 + x^4/2

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 20, normalized size = 1.05 \begin {gather*} \frac {x^{4}}{2} - 2 x^{3} - 20 x^{2} + 44 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4-6*x**3-40*x**2+44*x-1)/x,x)

[Out]

x**4/2 - 2*x**3 - 20*x**2 + 44*x - log(x)

________________________________________________________________________________________