3.23.46 \(\int \frac {4 x+10 x^2+12 x^3+4 x^4+(2+4 x+4 x^2) \log (x)}{x} \, dx\)

Optimal. Leaf size=11 \[ \left (2 x+x^2+\log (x)\right )^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 30, normalized size of antiderivative = 2.73, number of steps used = 8, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {14, 2357, 2295, 2301, 2304} \begin {gather*} x^4+4 x^3+4 x^2+2 x^2 \log (x)+\log ^2(x)+4 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x + 10*x^2 + 12*x^3 + 4*x^4 + (2 + 4*x + 4*x^2)*Log[x])/x,x]

[Out]

4*x^2 + 4*x^3 + x^4 + 4*x*Log[x] + 2*x^2*Log[x] + Log[x]^2

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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
]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 \left (2+5 x+6 x^2+2 x^3\right )+\frac {2 \left (1+2 x+2 x^2\right ) \log (x)}{x}\right ) \, dx\\ &=2 \int \left (2+5 x+6 x^2+2 x^3\right ) \, dx+2 \int \frac {\left (1+2 x+2 x^2\right ) \log (x)}{x} \, dx\\ &=4 x+5 x^2+4 x^3+x^4+2 \int \left (2 \log (x)+\frac {\log (x)}{x}+2 x \log (x)\right ) \, dx\\ &=4 x+5 x^2+4 x^3+x^4+2 \int \frac {\log (x)}{x} \, dx+4 \int \log (x) \, dx+4 \int x \log (x) \, dx\\ &=4 x^2+4 x^3+x^4+4 x \log (x)+2 x^2 \log (x)+\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 30, normalized size = 2.73 \begin {gather*} 4 x^2+4 x^3+x^4+4 x \log (x)+2 x^2 \log (x)+\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + 10*x^2 + 12*x^3 + 4*x^4 + (2 + 4*x + 4*x^2)*Log[x])/x,x]

[Out]

4*x^2 + 4*x^3 + x^4 + 4*x*Log[x] + 2*x^2*Log[x] + Log[x]^2

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fricas [B]  time = 0.68, size = 29, normalized size = 2.64 \begin {gather*} x^{4} + 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} \log \relax (x) + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+4*x+2)*log(x)+4*x^4+12*x^3+10*x^2+4*x)/x,x, algorithm="fricas")

[Out]

x^4 + 4*x^3 + 4*x^2 + 2*(x^2 + 2*x)*log(x) + log(x)^2

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giac [B]  time = 0.23, size = 29, normalized size = 2.64 \begin {gather*} x^{4} + 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} \log \relax (x) + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+4*x+2)*log(x)+4*x^4+12*x^3+10*x^2+4*x)/x,x, algorithm="giac")

[Out]

x^4 + 4*x^3 + 4*x^2 + 2*(x^2 + 2*x)*log(x) + log(x)^2

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maple [B]  time = 0.02, size = 31, normalized size = 2.82

method	result	size
default	\(x^{4}+2 x^{2} \ln \relax (x )+4 x^{2}+4 x^{3}+4 x \ln \relax (x )+\ln \relax (x )^{2}\)	\(31\)
norman	\(x^{4}+2 x^{2} \ln \relax (x )+4 x^{2}+4 x^{3}+4 x \ln \relax (x )+\ln \relax (x )^{2}\)	\(31\)
risch	\(\ln \relax (x )^{2}+\left (2 x^{2}+4 x \right ) \ln \relax (x )+x^{4}+4 x^{3}+4 x^{2}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+4*x+2)*ln(x)+4*x^4+12*x^3+10*x^2+4*x)/x,x,method=_RETURNVERBOSE)

[Out]

x^4+2*x^2*ln(x)+4*x^2+4*x^3+4*x*ln(x)+ln(x)^2

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maxima [B]  time = 0.36, size = 30, normalized size = 2.73 \begin {gather*} x^{4} + 4 \, x^{3} + 2 \, x^{2} \log \relax (x) + 4 \, x^{2} + 4 \, x \log \relax (x) + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+4*x+2)*log(x)+4*x^4+12*x^3+10*x^2+4*x)/x,x, algorithm="maxima")

[Out]

x^4 + 4*x^3 + 2*x^2*log(x) + 4*x^2 + 4*x*log(x) + log(x)^2

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mupad [B]  time = 1.39, size = 11, normalized size = 1.00 \begin {gather*} {\left (2\,x+\ln \relax (x)+x^2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log(x)*(4*x + 4*x^2 + 2) + 10*x^2 + 12*x^3 + 4*x^4)/x,x)

[Out]

(2*x + log(x) + x^2)^2

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sympy [B]  time = 0.11, size = 29, normalized size = 2.64 \begin {gather*} x^{4} + 4 x^{3} + 4 x^{2} + \left (2 x^{2} + 4 x\right ) \log {\relax (x )} + \log {\relax (x )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+4*x+2)*ln(x)+4*x**4+12*x**3+10*x**2+4*x)/x,x)

[Out]

x**4 + 4*x**3 + 4*x**2 + (2*x**2 + 4*x)*log(x) + log(x)**2

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