3.64.67 \(\int (1+e^x (3+x) \log (5)) \, dx\)

Optimal. Leaf size=12 \[ -3+x+e^x (2+x) \log (5) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} x-e^x \log (5)+e^x (x+3) \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^x*(3 + x)*Log[5],x]

[Out]

x - E^x*Log[5] + E^x*(3 + x)*Log[5]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\log (5) \int e^x (3+x) \, dx\\ &=x+e^x (3+x) \log (5)-\log (5) \int e^x \, dx\\ &=x-e^x \log (5)+e^x (3+x) \log (5)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.92 \begin {gather*} x+e^x (2+x) \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^x*(3 + x)*Log[5],x]

[Out]

x + E^x*(2 + x)*Log[5]

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fricas [A]  time = 0.57, size = 10, normalized size = 0.83 \begin {gather*} {\left (x + 2\right )} e^{x} \log \relax (5) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)*log(5)*exp(x)+1,x, algorithm="fricas")

[Out]

(x + 2)*e^x*log(5) + x

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giac [A]  time = 0.15, size = 10, normalized size = 0.83 \begin {gather*} {\left (x + 2\right )} e^{x} \log \relax (5) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)*log(5)*exp(x)+1,x, algorithm="giac")

[Out]

(x + 2)*e^x*log(5) + x

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maple [A]  time = 0.03, size = 11, normalized size = 0.92

method	result	size
risch	\({\mathrm e}^{x} \left (2+x \right ) \ln \relax (5)+x\)	\(11\)
default	\(x +x \,{\mathrm e}^{x} \ln \relax (5)+2 \,{\mathrm e}^{x} \ln \relax (5)\)	\(15\)
norman	\(x +x \,{\mathrm e}^{x} \ln \relax (5)+2 \,{\mathrm e}^{x} \ln \relax (5)\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)*ln(5)*exp(x)+1,x,method=_RETURNVERBOSE)

[Out]

exp(x)*(2+x)*ln(5)+x

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maxima [A]  time = 0.37, size = 16, normalized size = 1.33 \begin {gather*} {\left ({\left (x - 1\right )} e^{x} + 3 \, e^{x}\right )} \log \relax (5) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)*log(5)*exp(x)+1,x, algorithm="maxima")

[Out]

((x - 1)*e^x + 3*e^x)*log(5) + x

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mupad [B]  time = 0.06, size = 14, normalized size = 1.17 \begin {gather*} x+2\,{\mathrm {e}}^x\,\ln \relax (5)+x\,{\mathrm {e}}^x\,\ln \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*log(5)*(x + 3) + 1,x)

[Out]

x + 2*exp(x)*log(5) + x*exp(x)*log(5)

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sympy [A]  time = 0.09, size = 14, normalized size = 1.17 \begin {gather*} x + \left (x \log {\relax (5 )} + 2 \log {\relax (5 )}\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)*ln(5)*exp(x)+1,x)

[Out]

x + (x*log(5) + 2*log(5))*exp(x)

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