### 3.47 $$\int 5^x x \, dx$$

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

[Out]

-(5^x/Log[5]^2) + (5^x*x)/Log[5]

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Rubi [A]  time = 0.0085244, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {2176, 2194} $\frac{5^x x}{\log (5)}-\frac{5^x}{\log ^2(5)}$

Antiderivative was successfully veriﬁed.

[In]

Int[5^x*x,x]

[Out]

-(5^x/Log[5]^2) + (5^x*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{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*}

Mathematica [A]  time = 0.0028747, size = 14, normalized size = 0.74 $\frac{5^x (x \log (5)-1)}{\log ^2(5)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[5^x*x,x]

[Out]

(5^x*(-1 + x*Log[5]))/Log[5]^2

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \begin{align*}{\frac{ \left ( \ln \left ( 5 \right ) x-1 \right ){5}^{x}}{ \left ( \ln \left ( 5 \right ) \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(5^x*x,x)

[Out]

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

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Maxima [A]  time = 1.42486, size = 19, normalized size = 1. \begin{align*} \frac{{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \end{align*}

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 = 2.23962, size = 39, normalized size = 2.05 \begin{align*} \frac{{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \end{align*}

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 = 0.091439, size = 14, normalized size = 0.74 \begin{align*} \frac{5^{x} \left (x \log{\left (5 \right )} - 1\right )}{\log{\left (5 \right )}^{2}} \end{align*}

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 = 1.04894, size = 19, normalized size = 1. \begin{align*} \frac{{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(5^x*x,x, algorithm="giac")

[Out]

(x*log(5) - 1)*5^x/log(5)^2