3.8.72 \(\int e^5 (320 x^3+80 x^4) \log (5) \, dx\)

Optimal. Leaf size=13 \[ 16 e^5 x^4 (5+x) \log (5) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.62, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12} \begin {gather*} 16 e^5 x^5 \log (5)+80 e^5 x^4 \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^5*(320*x^3 + 80*x^4)*Log[5],x]

[Out]

80*E^5*x^4*Log[5] + 16*E^5*x^5*Log[5]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (e^5 \log (5)\right ) \int \left (320 x^3+80 x^4\right ) \, dx\\ &=80 e^5 x^4 \log (5)+16 e^5 x^5 \log (5)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 18, normalized size = 1.38 \begin {gather*} 80 e^5 \left (x^4+\frac {x^5}{5}\right ) \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^5*(320*x^3 + 80*x^4)*Log[5],x]

[Out]

80*E^5*(x^4 + x^5/5)*Log[5]

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 15, normalized size = 1.15 \begin {gather*} 16 \, {\left (x^{5} + 5 \, x^{4}\right )} e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((80*x^4+320*x^3)*exp(5)*log(5),x, algorithm="fricas")

[Out]

16*(x^5 + 5*x^4)*e^5*log(5)

________________________________________________________________________________________

giac [A]  time = 0.51, size = 15, normalized size = 1.15 \begin {gather*} 16 \, {\left (x^{5} + 5 \, x^{4}\right )} e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((80*x^4+320*x^3)*exp(5)*log(5),x, algorithm="giac")

[Out]

16*(x^5 + 5*x^4)*e^5*log(5)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 13, normalized size = 1.00

method	result	size
gosper	\(16 \,{\mathrm e}^{5} \left (5+x \right ) x^{4} \ln \relax (5)\)	\(13\)
default	\({\mathrm e}^{5} \ln \relax (5) \left (16 x^{5}+80 x^{4}\right )\)	\(17\)
norman	\(80 x^{4} {\mathrm e}^{5} \ln \relax (5)+16 x^{5} {\mathrm e}^{5} \ln \relax (5)\)	\(20\)
risch	\(80 x^{4} {\mathrm e}^{5} \ln \relax (5)+16 x^{5} {\mathrm e}^{5} \ln \relax (5)\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80*x^4+320*x^3)*exp(5)*ln(5),x,method=_RETURNVERBOSE)

[Out]

16*exp(5)*(5+x)*x^4*ln(5)

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 15, normalized size = 1.15 \begin {gather*} 16 \, {\left (x^{5} + 5 \, x^{4}\right )} e^{5} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((80*x^4+320*x^3)*exp(5)*log(5),x, algorithm="maxima")

[Out]

16*(x^5 + 5*x^4)*e^5*log(5)

________________________________________________________________________________________

mupad [B]  time = 0.58, size = 12, normalized size = 0.92 \begin {gather*} 16\,x^4\,{\mathrm {e}}^5\,\ln \relax (5)\,\left (x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5)*log(5)*(320*x^3 + 80*x^4),x)

[Out]

16*x^4*exp(5)*log(5)*(x + 5)

________________________________________________________________________________________

sympy [A]  time = 0.05, size = 22, normalized size = 1.69 \begin {gather*} 16 x^{5} e^{5} \log {\relax (5 )} + 80 x^{4} e^{5} \log {\relax (5 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((80*x**4+320*x**3)*exp(5)*ln(5),x)

[Out]

16*x**5*exp(5)*log(5) + 80*x**4*exp(5)*log(5)

________________________________________________________________________________________