3.11.18 \(\int \frac {2+72 e^4+18 e^4 \log (3+x)}{e^4 (27+9 x)} \, dx\)

Optimal. Leaf size=21 \[ -e^2+\left (4+\frac {1}{9 e^4}+\log (3+x)\right )^2 \]

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.19, 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, 2390, 2301} \begin {gather*} \frac {\left (9 e^4 \log (x+3)+36 e^4+1\right )^2}{81 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 72*E^4 + 18*E^4*Log[3 + x])/(E^4*(27 + 9*x)),x]

[Out]

(1 + 36*E^4 + 9*E^4*Log[3 + x])^2/(81*E^8)

Rule 12

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2+72 e^4+18 e^4 \log (3+x)}{27+9 x} \, dx}{e^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {2+72 e^4+18 e^4 \log (x)}{9 x} \, dx,x,3+x\right )}{e^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {2+72 e^4+18 e^4 \log (x)}{x} \, dx,x,3+x\right )}{9 e^4}\\ &=\frac {\left (1+36 e^4+9 e^4 \log (3+x)\right )^2}{81 e^8}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.19 \begin {gather*} \frac {\left (1+36 e^4+9 e^4 \log (3+x)\right )^2}{81 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 72*E^4 + 18*E^4*Log[3 + x])/(E^4*(27 + 9*x)),x]

[Out]

(1 + 36*E^4 + 9*E^4*Log[3 + x])^2/(81*E^8)

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fricas [A]  time = 0.93, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{9} \, {\left (9 \, e^{4} \log \left (x + 3\right )^{2} + 2 \, {\left (36 \, e^{4} + 1\right )} \log \left (x + 3\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(1)^4*log(3+x)+72*exp(1)^4+2)/(9*x+27)/exp(1)^4,x, algorithm="fricas")

[Out]

1/9*(9*e^4*log(x + 3)^2 + 2*(36*e^4 + 1)*log(x + 3))*e^(-4)

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giac [A]  time = 0.37, size = 29, normalized size = 1.38 \begin {gather*} \frac {1}{9} \, {\left (9 \, e^{4} \log \left (x + 3\right )^{2} + 72 \, e^{4} \log \left (x + 3\right ) + 2 \, \log \left (x + 3\right )\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(1)^4*log(3+x)+72*exp(1)^4+2)/(9*x+27)/exp(1)^4,x, algorithm="giac")

[Out]

1/9*(9*e^4*log(x + 3)^2 + 72*e^4*log(x + 3) + 2*log(x + 3))*e^(-4)

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maple [A]  time = 0.16, size = 26, normalized size = 1.24

method	result	size
risch	\(\ln \left (3+x \right )^{2}+8 \,{\mathrm e}^{-4} \ln \left (3+x \right ) {\mathrm e}^{4}+\frac {2 \,{\mathrm e}^{-4} \ln \left (3+x \right )}{9}\)	\(26\)
derivativedivides	\(\frac {2 \,{\mathrm e}^{-4} \left (\frac {9 \ln \left (3+x \right )^{2} {\mathrm e}^{4}}{2}+36 \,{\mathrm e}^{4} \ln \left (3+x \right )+\ln \left (3+x \right )\right )}{9}\)	\(34\)
default	\(\frac {2 \,{\mathrm e}^{-4} \left (\frac {9 \ln \left (3+x \right )^{2} {\mathrm e}^{4}}{2}+36 \,{\mathrm e}^{4} \ln \left (3+x \right )+\ln \left (3+x \right )\right )}{9}\)	\(34\)
norman	\(\left (\frac {2 \left (36 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-1} \ln \left (3+x \right )}{9}+{\mathrm e}^{3} \ln \left (3+x \right )^{2}\right ) {\mathrm e}^{-3}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*exp(1)^4*ln(3+x)+72*exp(1)^4+2)/(9*x+27)/exp(1)^4,x,method=_RETURNVERBOSE)

[Out]

ln(3+x)^2+8*exp(-4)*ln(3+x)*exp(4)+2/9*exp(-4)*ln(3+x)

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maxima [A]  time = 0.79, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{81} \, {\left (9 \, e^{4} \log \left (x + 3\right ) + 36 \, e^{4} + 1\right )}^{2} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(1)^4*log(3+x)+72*exp(1)^4+2)/(9*x+27)/exp(1)^4,x, algorithm="maxima")

[Out]

1/81*(9*e^4*log(x + 3) + 36*e^4 + 1)^2*e^(-8)

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mupad [B]  time = 0.19, size = 22, normalized size = 1.05 \begin {gather*} \frac {\ln \left (x+3\right )\,{\mathrm {e}}^{-4}\,\left (72\,{\mathrm {e}}^4+9\,\ln \left (x+3\right )\,{\mathrm {e}}^4+2\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*(72*exp(4) + 18*log(x + 3)*exp(4) + 2))/(9*x + 27),x)

[Out]

(log(x + 3)*exp(-4)*(72*exp(4) + 9*log(x + 3)*exp(4) + 2))/9

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sympy [A]  time = 0.12, size = 31, normalized size = 1.48 \begin {gather*} \log {\left (x + 3 \right )}^{2} + \frac {\left (2 + 72 e^{4}\right ) \log {\left (9 x e^{4} + 27 e^{4} \right )}}{9 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(1)**4*ln(3+x)+72*exp(1)**4+2)/(9*x+27)/exp(1)**4,x)

[Out]

log(x + 3)**2 + (2 + 72*exp(4))*exp(-4)*log(9*x*exp(4) + 27*exp(4))/9

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