3.52.74 \(\int \frac {27 e^{81 e^2+81 x} x+e^{1+\log ^2(x)} x^2 (2+2 \log (x))}{x} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{3} e^{81 \left (e^2+x\right )}+e^{(1+\log (x))^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {14, 2194, 2288} \begin {gather*} x^2 e^{\log ^2(x)+1}+\frac {1}{3} e^{81 x+81 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(27*E^(81*E^2 + 81*x)*x + E^(1 + Log[x]^2)*x^2*(2 + 2*Log[x]))/x,x]

[Out]

E^(81*E^2 + 81*x)/3 + E^(1 + Log[x]^2)*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 2194

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (27 e^{81 e^2+81 x}+2 e^{1+\log ^2(x)} x (1+\log (x))\right ) \, dx\\ &=2 \int e^{1+\log ^2(x)} x (1+\log (x)) \, dx+27 \int e^{81 e^2+81 x} \, dx\\ &=\frac {1}{3} e^{81 e^2+81 x}+e^{1+\log ^2(x)} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.27 \begin {gather*} \frac {1}{3} e^{81 e^2+81 x}+e^{1+\log ^2(x)} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(27*E^(81*E^2 + 81*x)*x + E^(1 + Log[x]^2)*x^2*(2 + 2*Log[x]))/x,x]

[Out]

E^(81*E^2 + 81*x)/3 + E^(1 + Log[x]^2)*x^2

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fricas [A]  time = 0.54, size = 23, normalized size = 1.05 \begin {gather*} e^{\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} + \frac {1}{3} \, e^{\left (81 \, x + 81 \, e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)+2)*exp(log(x)^2+2*log(x)+1)+27*x*exp(81*exp(2)+81*x))/x,x, algorithm="fricas")

[Out]

e^(log(x)^2 + 2*log(x) + 1) + 1/3*e^(81*x + 81*e^2)

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giac [A]  time = 0.21, size = 23, normalized size = 1.05 \begin {gather*} x^{2} e^{\left (\log \relax (x)^{2} + 1\right )} + \frac {1}{3} \, e^{\left (81 \, x + 81 \, e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)+2)*exp(log(x)^2+2*log(x)+1)+27*x*exp(81*exp(2)+81*x))/x,x, algorithm="giac")

[Out]

x^2*e^(log(x)^2 + 1) + 1/3*e^(81*x + 81*e^2)

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maple [A]  time = 0.05, size = 24, normalized size = 1.09

method	result	size
default	\({\mathrm e}^{\ln \relax (x )^{2}+2 \ln \relax (x )+1}+\frac {{\mathrm e}^{81 \,{\mathrm e}^{2}+81 x}}{3}\)	\(24\)
norman	\({\mathrm e}^{\ln \relax (x )^{2}+2 \ln \relax (x )+1}+\frac {{\mathrm e}^{81 \,{\mathrm e}^{2}+81 x}}{3}\)	\(24\)
risch	\(x^{2} {\mathrm e}^{\ln \relax (x )^{2}+1}+\frac {{\mathrm e}^{81 \,{\mathrm e}^{2}+81 x}}{3}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(x)+2)*exp(ln(x)^2+2*ln(x)+1)+27*x*exp(81*exp(2)+81*x))/x,x,method=_RETURNVERBOSE)

[Out]

exp(ln(x)^2+2*ln(x)+1)+1/3*exp(81*exp(2)+81*x)

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maxima [C]  time = 0.40, size = 63, normalized size = 2.86 \begin {gather*} -\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-{\left (\log \relax (x) + 1\right )}^{2}}\right ) - 1\right )} {\left (\log \relax (x) + 1\right )}}{\sqrt {-{\left (\log \relax (x) + 1\right )}^{2}}} - i \, \sqrt {\pi } \operatorname {erf}\left (i \, \log \relax (x) + i\right ) + e^{\left ({\left (\log \relax (x) + 1\right )}^{2}\right )} + \frac {1}{3} \, e^{\left (81 \, x + 81 \, e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)+2)*exp(log(x)^2+2*log(x)+1)+27*x*exp(81*exp(2)+81*x))/x,x, algorithm="maxima")

[Out]

-sqrt(pi)*(erf(sqrt(-(log(x) + 1)^2)) - 1)*(log(x) + 1)/sqrt(-(log(x) + 1)^2) - I*sqrt(pi)*erf(I*log(x) + I) +
e^((log(x) + 1)^2) + 1/3*e^(81*x + 81*e^2)

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mupad [B]  time = 3.31, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\mathrm {e}}^{81\,{\mathrm {e}}^2}\,{\mathrm {e}}^{81\,x}}{3}+x^2\,\mathrm {e}\,{\mathrm {e}}^{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x*exp(81*x + 81*exp(2)) + exp(2*log(x) + log(x)^2 + 1)*(2*log(x) + 2))/x,x)

[Out]

(exp(81*exp(2))*exp(81*x))/3 + x^2*exp(1)*exp(log(x)^2)

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sympy [A]  time = 0.37, size = 22, normalized size = 1.00 \begin {gather*} x^{2} e^{\log {\relax (x )}^{2} + 1} + \frac {e^{81 x + 81 e^{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(x)+2)*exp(ln(x)**2+2*ln(x)+1)+27*x*exp(81*exp(2)+81*x))/x,x)

[Out]

x**2*exp(log(x)**2 + 1) + exp(81*x + 81*exp(2))/3

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