3.72.45 \(\int \frac {1}{3} (24+e^{\frac {1}{3} (9+5 x)} (3+5 x)+3 \log (4)) \, dx\)

Optimal. Leaf size=19 \[ -x+x \left (9+e^{3+\frac {5 x}{3}}+\log (4)\right ) \]

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Rubi [B]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2176, 2194} \begin {gather*} -\frac {3}{5} e^{\frac {1}{3} (5 x+9)}+\frac {1}{5} e^{\frac {1}{3} (5 x+9)} (5 x+3)+x (8+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24 + E^((9 + 5*x)/3)*(3 + 5*x) + 3*Log[4])/3,x]

[Out]

(-3*E^((9 + 5*x)/3))/5 + (E^((9 + 5*x)/3)*(3 + 5*x))/5 + x*(8 + Log[4])

Rule 12

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

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (24+e^{\frac {1}{3} (9+5 x)} (3+5 x)+3 \log (4)\right ) \, dx\\ &=x (8+\log (4))+\frac {1}{3} \int e^{\frac {1}{3} (9+5 x)} (3+5 x) \, dx\\ &=\frac {1}{5} e^{\frac {1}{3} (9+5 x)} (3+5 x)+x (8+\log (4))-\int e^{\frac {1}{3} (9+5 x)} \, dx\\ &=-\frac {3}{5} e^{\frac {1}{3} (9+5 x)}+\frac {1}{5} e^{\frac {1}{3} (9+5 x)} (3+5 x)+x (8+\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 1.26 \begin {gather*} \frac {1}{3} \left (24 x+3 e^{3+\frac {5 x}{3}} x+x \log (64)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 + E^((9 + 5*x)/3)*(3 + 5*x) + 3*Log[4])/3,x]

[Out]

(24*x + 3*E^(3 + (5*x)/3)*x + x*Log[64])/3

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fricas [A]  time = 0.80, size = 17, normalized size = 0.89 \begin {gather*} x e^{\left (\frac {5}{3} \, x + 3\right )} + 2 \, x \log \relax (2) + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(5*x+3)*exp(5/3*x+3)+2*log(2)+8,x, algorithm="fricas")

[Out]

x*e^(5/3*x + 3) + 2*x*log(2) + 8*x

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giac [A]  time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} x e^{\left (\frac {5}{3} \, x + 3\right )} + 2 \, x \log \relax (2) + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(5*x+3)*exp(5/3*x+3)+2*log(2)+8,x, algorithm="giac")

[Out]

x*e^(5/3*x + 3) + 2*x*log(2) + 8*x

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maple [A]  time = 0.03, size = 18, normalized size = 0.95

method	result	size
norman	\(\left (8+2 \ln \relax (2)\right ) x +{\mathrm e}^{\frac {5 x}{3}+3} x\)	\(18\)
risch	\({\mathrm e}^{\frac {5 x}{3}+3} x +2 x \ln \relax (2)+8 x\)	\(18\)
default	\(8 x +\frac {3 \,{\mathrm e}^{\frac {5 x}{3}+3} \left (\frac {5 x}{3}+3\right )}{5}-\frac {9 \,{\mathrm e}^{\frac {5 x}{3}+3}}{5}+2 x \ln \relax (2)\)	\(31\)
derivativedivides	\(8 x +\frac {72}{5}+\frac {3 \,{\mathrm e}^{\frac {5 x}{3}+3} \left (\frac {5 x}{3}+3\right )}{5}-\frac {9 \,{\mathrm e}^{\frac {5 x}{3}+3}}{5}+\frac {6 \ln \relax (2) \left (\frac {5 x}{3}+3\right )}{5}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(5*x+3)*exp(5/3*x+3)+2*ln(2)+8,x,method=_RETURNVERBOSE)

[Out]

(8+2*ln(2))*x+exp(5/3*x+3)*x

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maxima [A]  time = 0.38, size = 17, normalized size = 0.89 \begin {gather*} x e^{\left (\frac {5}{3} \, x + 3\right )} + 2 \, x \log \relax (2) + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(5*x+3)*exp(5/3*x+3)+2*log(2)+8,x, algorithm="maxima")

[Out]

x*e^(5/3*x + 3) + 2*x*log(2) + 8*x

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mupad [B]  time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} x\,\left (\frac {\ln \left (64\right )}{3}+8\right )+x\,{\mathrm {e}}^{\frac {5\,x}{3}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*log(2) + (exp((5*x)/3 + 3)*(5*x + 3))/3 + 8,x)

[Out]

x*(log(64)/3 + 8) + x*exp((5*x)/3 + 3)

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sympy [A]  time = 0.10, size = 17, normalized size = 0.89 \begin {gather*} x e^{\frac {5 x}{3} + 3} + x \left (2 \log {\relax (2 )} + 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(5*x+3)*exp(5/3*x+3)+2*ln(2)+8,x)

[Out]

x*exp(5*x/3 + 3) + x*(2*log(2) + 8)

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