3.54.56 \(\int \frac {e^{3+x^3}+3 e^{3+x^3} x^3 \log (x)}{x} \, dx\)

Optimal. Leaf size=10 \[ e^{3+x^3} \log (x) \]

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Rubi [A]  time = 0.09, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {14, 2210, 2209, 2554, 12} \begin {gather*} e^{x^3+3} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3 + x^3)*Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2554

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} e^{3+x^3} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3 + x^3)*Log[x]

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fricas [A]  time = 0.76, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x^{3} + 3\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^3 + 3)*log(x)

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giac [A]  time = 0.20, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x^{3} + 3\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^3 + 3)*log(x)

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maple [A]  time = 0.04, size = 10, normalized size = 1.00

method	result	size
norman	\({\mathrm e}^{x^{3}+3} \ln \relax (x )\)	\(10\)
risch	\({\mathrm e}^{x^{3}+3} \ln \relax (x )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.40, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x^{3} + 3\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^3 + 3)*log(x)

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mupad [B]  time = 3.59, size = 9, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{x^3}\,{\mathrm {e}}^3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^3 + 3) + 3*x^3*exp(x^3 + 3)*log(x))/x,x)

[Out]

exp(x^3)*exp(3)*log(x)

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sympy [A]  time = 0.27, size = 8, normalized size = 0.80 \begin {gather*} e^{x^{3} + 3} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3*exp(x**3+3)*ln(x)+exp(x**3+3))/x,x)

[Out]

exp(x**3 + 3)*log(x)

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