3.1.47 \(\int 5^x x \, dx\) [47]

Optimal. Leaf size=19 \[ -\frac {5^x}{\log ^2(5)}+\frac {5^x x}{\log (5)} \]

[Out]

-5^x/ln(5)^2+5^x*x/ln(5)

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2207, 2225} \begin {gather*} \frac {5^x x}{\log (5)}-\frac {5^x}{\log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[5^x*x,x]

[Out]

-(5^x/Log[5]^2) + (5^x*x)/Log[5]

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 {align*} \int 5^x x \, dx &=\frac {5^x x}{\log (5)}-\frac {\int 5^x \, dx}{\log (5)}\\ &=-\frac {5^x}{\log ^2(5)}+\frac {5^x x}{\log (5)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.74 \begin {gather*} \frac {5^x (-1+x \log (5))}{\log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5^x*x,x]

[Out]

(5^x*(-1 + x*Log[5]))/Log[5]^2

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Mathics [A]
time = 1.79, size = 14, normalized size = 0.74 \begin {gather*} \frac {\left (-1+x \text {Log}\left [5\right ]\right ) 5^x}{\text {Log}\left [5\right ]^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*5^x,x]')

[Out]

(-1 + x Log[5]) 5 ^ x / Log[5] ^ 2

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Maple [A]
time = 0.02, size = 15, normalized size = 0.79

method	result	size
gosper	\(\frac {\left (x \ln \left (5\right )-1\right ) 5^{x}}{\ln \left (5\right )^{2}}\)	\(15\)
risch	\(\frac {\left (x \ln \left (5\right )-1\right ) 5^{x}}{\ln \left (5\right )^{2}}\)	\(15\)
meijerg	\(\frac {1-\frac {\left (2-2 x \ln \left (5\right )\right ) {\mathrm e}^{x \ln \left (5\right )}}{2}}{\ln \left (5\right )^{2}}\)	\(22\)
norman	\(\frac {x \,{\mathrm e}^{x \ln \left (5\right )}}{\ln \left (5\right )}-\frac {{\mathrm e}^{x \ln \left (5\right )}}{\ln \left (5\right )^{2}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5^x*x,x,method=_RETURNVERBOSE)

[Out]

(x*ln(5)-1)*5^x/ln(5)^2

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Maxima [A]
time = 0.36, size = 14, normalized size = 0.74 \begin {gather*} \frac {{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^x*x,x, algorithm="maxima")

[Out]

(x*log(5) - 1)*5^x/log(5)^2

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Fricas [A]
time = 0.33, size = 14, normalized size = 0.74 \begin {gather*} \frac {{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^x*x,x, algorithm="fricas")

[Out]

(x*log(5) - 1)*5^x/log(5)^2

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Sympy [A]
time = 0.05, size = 14, normalized size = 0.74 \begin {gather*} \frac {5^{x} \left (x \log {\left (5 \right )} - 1\right )}{\log {\left (5 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5**x*x,x)

[Out]

5**x*(x*log(5) - 1)/log(5)**2

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Giac [A]
time = 0.00, size = 17, normalized size = 0.89 \begin {gather*} \frac {\left (x \ln \left (5\right )-1\right ) \mathrm {e}^{x \ln \left (5\right )}}{\ln ^{2}\left (5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^x*x,x)

[Out]

(x*log(5) - 1)*5^x/log(5)^2

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.74 \begin {gather*} \frac {5^x\,\left (x\,\ln \left (5\right )-1\right )}{{\ln \left (5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5^x*x,x)

[Out]

(5^x*(x*log(5) - 1))/log(5)^2

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