3.23.51 \(\int \frac {4+x^2+x^4+3 x^4 \log (x)}{x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {-4+x+x^2+x^4 \log (x)}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {14, 2304} \begin {gather*} x^3 \log (x)+x-\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + x^2 + x^4 + 3*x^4*Log[x])/x^2,x]

[Out]

-4/x + x + x^3*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]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} -\frac {4}{x}+x+x^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + x^2 + x^4 + 3*x^4*Log[x])/x^2,x]

[Out]

-4/x + x + x^3*Log[x]

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fricas [A]  time = 1.05, size = 15, normalized size = 0.94 \begin {gather*} \frac {x^{4} \log \relax (x) + x^{2} - 4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4*log(x) + x^2 - 4)/x

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giac [A]  time = 0.33, size = 13, normalized size = 0.81 \begin {gather*} x^{3} \log \relax (x) + x - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*log(x) + x - 4/x

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maple [A]  time = 0.02, size = 14, normalized size = 0.88

method	result	size
default	\(x^{3} \ln \relax (x )+x -\frac {4}{x}\)	\(14\)
norman	\(\frac {-4+x^{2}+x^{4} \ln \relax (x )}{x}\)	\(16\)
risch	\(x^{3} \ln \relax (x )+\frac {x^{2}-4}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*ln(x)+x-4/x

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maxima [A]  time = 0.51, size = 13, normalized size = 0.81 \begin {gather*} x^{3} \log \relax (x) + x - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*log(x) + x - 4/x

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mupad [B]  time = 1.30, size = 16, normalized size = 1.00 \begin {gather*} x\,\left (x^2\,\ln \relax (x)+1\right )-\frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4*log(x) + x^2 + x^4 + 4)/x^2,x)

[Out]

x*(x^2*log(x) + 1) - 4/x

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sympy [A]  time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} x^{3} \log {\relax (x )} + x - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*log(x) + x - 4/x

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