3.70.4 \(\int (1+2 x^2 \log (x)+3 x^2 \log (e^{\log ^2(x)} (28+\log (3)))) \, dx\)

Optimal. Leaf size=18 \[ x+x^3 \log \left (e^{\log ^2(x)} (28+\log (3))\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2304, 30, 2555, 12} \begin {gather*} x^3 \log \left ((28+\log (3)) e^{\log ^2(x)}\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 2*x^2*Log[x] + 3*x^2*Log[E^Log[x]^2*(28 + Log[3])],x]

[Out]

x + x^3*Log[E^Log[x]^2*(28 + Log[3])]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2555

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify
[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+2 \int x^2 \log (x) \, dx+3 \int x^2 \log \left (e^{\log ^2(x)} (28+\log (3))\right ) \, dx\\ &=x-\frac {2 x^3}{9}+\frac {2}{3} x^3 \log (x)+x^3 \log \left (e^{\log ^2(x)} (28+\log (3))\right )-3 \int \frac {2}{3} x^2 \log (x) \, dx\\ &=x-\frac {2 x^3}{9}+\frac {2}{3} x^3 \log (x)+x^3 \log \left (e^{\log ^2(x)} (28+\log (3))\right )-2 \int x^2 \log (x) \, dx\\ &=x+x^3 \log \left (e^{\log ^2(x)} (28+\log (3))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} x+x^3 \log \left (e^{\log ^2(x)} (28+\log (3))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 2*x^2*Log[x] + 3*x^2*Log[E^Log[x]^2*(28 + Log[3])],x]

[Out]

x + x^3*Log[E^Log[x]^2*(28 + Log[3])]

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fricas [A]  time = 0.58, size = 19, normalized size = 1.06 \begin {gather*} x^{3} \log \relax (x)^{2} + x^{3} \log \left (\log \relax (3) + 28\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*log((log(3)+28)*exp(log(x)^2))+2*x^2*log(x)+1,x, algorithm="fricas")

[Out]

x^3*log(x)^2 + x^3*log(log(3) + 28) + x

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giac [A]  time = 0.26, size = 19, normalized size = 1.06 \begin {gather*} x^{3} \log \relax (x)^{2} + x^{3} \log \left (\log \relax (3) + 28\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*log((log(3)+28)*exp(log(x)^2))+2*x^2*log(x)+1,x, algorithm="giac")

[Out]

x^3*log(x)^2 + x^3*log(log(3) + 28) + x

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maple [A]  time = 0.10, size = 22, normalized size = 1.22

method	result	size
risch	\(x^{3} \ln \left ({\mathrm e}^{\ln \relax (x )^{2}}\right )+\ln \left (\ln \relax (3)+28\right ) x^{3}+x\)	\(22\)
default	\(x +x^{3} \ln \relax (x )^{2}+\left (\ln \left (\left (\ln \relax (3)+28\right ) {\mathrm e}^{\ln \relax (x )^{2}}\right )-\ln \relax (x )^{2}\right ) x^{3}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2*ln((ln(3)+28)*exp(ln(x)^2))+2*x^2*ln(x)+1,x,method=_RETURNVERBOSE)

[Out]

x^3*ln(exp(ln(x)^2))+ln(ln(3)+28)*x^3+x

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maxima [B]  time = 0.36, size = 40, normalized size = 2.22 \begin {gather*} -\frac {2}{9} \, x^{3} {\left (3 \, \log \relax (x) - 1\right )} + x^{3} \log \left ({\left (\log \relax (3) + 28\right )} e^{\left (\log \relax (x)^{2}\right )}\right ) + \frac {2}{3} \, x^{3} \log \relax (x) - \frac {2}{9} \, x^{3} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*log((log(3)+28)*exp(log(x)^2))+2*x^2*log(x)+1,x, algorithm="maxima")

[Out]

-2/9*x^3*(3*log(x) - 1) + x^3*log((log(3) + 28)*e^(log(x)^2)) + 2/3*x^3*log(x) - 2/9*x^3 + x

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mupad [B]  time = 4.29, size = 21, normalized size = 1.17 \begin {gather*} x+x^3\,\ln \left ({\mathrm {e}}^{{\ln \relax (x)}^2}\right )+x^3\,\ln \left (\ln \relax (3)+28\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x^2*log(x) + 3*x^2*log(exp(log(x)^2)*(log(3) + 28)) + 1,x)

[Out]

x + x^3*log(exp(log(x)^2)) + x^3*log(log(3) + 28)

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sympy [A]  time = 0.26, size = 19, normalized size = 1.06 \begin {gather*} x^{3} \log {\relax (x )}^{2} + x^{3} \log {\left (\log {\relax (3 )} + 28 \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x**2*ln((ln(3)+28)*exp(ln(x)**2))+2*x**2*ln(x)+1,x)

[Out]

x**3*log(x)**2 + x**3*log(log(3) + 28) + x

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