3.97.45 \(\int \frac {1+5^x x \log (5)}{x} \, dx\)

Optimal. Leaf size=8 \[ 5^x+\log (2 x) \]

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Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {14, 2194} \begin {gather*} 5^x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 5^x*x*Log[5])/x,x]

[Out]

5^x + Log[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 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} &=\int \left (\frac {1}{x}+5^x \log (5)\right ) \, dx\\ &=\log (x)+\log (5) \int 5^x \, dx\\ &=5^x+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.75 \begin {gather*} 5^x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 5^x*x*Log[5])/x,x]

[Out]

5^x + Log[x]

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fricas [A]  time = 0.74, size = 6, normalized size = 0.75 \begin {gather*} 5^{x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^x + log(x)

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giac [A]  time = 0.15, size = 7, normalized size = 0.88 \begin {gather*} 5^{x} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^x + log(abs(x))

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maple [A]  time = 0.04, size = 7, normalized size = 0.88

method	result	size
risch	\(5^{x}+\ln \relax (x )\)	\(7\)
norman	\({\mathrm e}^{x \ln \relax (5)}+\ln \relax (x )\)	\(9\)
derivativedivides	\(\ln \left (x \ln \relax (5)\right )+{\mathrm e}^{x \ln \relax (5)}\)	\(12\)
default	\(\ln \left (x \ln \relax (5)\right )+{\mathrm e}^{x \ln \relax (5)}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^x+ln(x)

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maxima [A]  time = 0.35, size = 6, normalized size = 0.75 \begin {gather*} 5^{x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5^x + log(x)

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mupad [B]  time = 5.42, size = 6, normalized size = 0.75 \begin {gather*} \ln \relax (x)+5^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x*log(5))*log(5) + 1)/x,x)

[Out]

log(x) + 5^x

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sympy [A]  time = 0.09, size = 8, normalized size = 1.00 \begin {gather*} e^{x \log {\relax (5 )}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*log(5)) + log(x)

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