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 = 0.017137, 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 \[ \frac{5^x x}{\log (5)}-\frac{5^x}{\log ^2(5)} \]

Antiderivative was successfully verified.

[In]  Int[5^x*x,x]

[Out]

-(5^x/Log[5]^2) + (5^x*x)/Log[5]

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Rubi in Sympy [A]  time = 1.03444, size = 15, normalized size = 0.79 \[ \frac{5^{x} x}{\log{\left (5 \right )}} - \frac{5^{x}}{\log{\left (5 \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(5**x*x,x)

[Out]

5**x*x/log(5) - 5**x/log(5)**2

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Mathematica [A]  time = 0.00446792, 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 = 0.01, size = 15, normalized size = 0.8 \[{\frac{ \left ( \ln \left ( 5 \right ) x-1 \right ){5}^{x}}{ \left ( \ln \left ( 5 \right ) \right ) ^{2}}} \]

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 = 1.50922, size = 19, normalized size = 1. \[ \frac{{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \]

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 = 0.249188, size = 19, normalized size = 1. \[ \frac{{\left (x \log \left (5\right ) - 1\right )} 5^{x}}{\log \left (5\right )^{2}} \]

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 = 0.087708, size = 14, normalized size = 0.74 \[ \frac{5^{x} \left (x \log{\left (5 \right )} - 1\right )}{\log{\left (5 \right )}^{2}} \]

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/XCAS [A]  time = 0.198895, size = 22, normalized size = 1.16 \[ \frac{{\left (x{\rm ln}\left (5\right ) - 1\right )} e^{\left (x{\rm ln}\left (5\right )\right )}}{{\rm ln}\left (5\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(5^x*x,x, algorithm="giac")

[Out]

(x*ln(5) - 1)*e^(x*ln(5))/ln(5)^2