3.33.19 \(\int \frac {7 e^4+e^{2 x} (6+12 x)+e^{2+x} (12+12 x)}{6 e^{2 x} x+12 e^{2+x} x+e^4 (22+7 x)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (-2+x+6 \left (4+\frac {\left (x+e^{-2+x} x\right )^2}{x}\right )\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6684} \begin {gather*} \log \left (6 e^{2 x} x+12 e^{x+2} x+e^4 (7 x+22)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7*E^4 + E^(2*x)*(6 + 12*x) + E^(2 + x)*(12 + 12*x))/(6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x)),x]

[Out]

Log[6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x)]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (6 e^{2 x} x+12 e^{2+x} x+e^4 (22+7 x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 29, normalized size = 1.26 \begin {gather*} \log \left (22 e^4+7 e^4 x+6 e^{2 x} x+12 e^{2+x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*E^4 + E^(2*x)*(6 + 12*x) + E^(2 + x)*(12 + 12*x))/(6*E^(2*x)*x + 12*E^(2 + x)*x + E^4*(22 + 7*x))
,x]

[Out]

Log[22*E^4 + 7*E^4*x + 6*E^(2*x)*x + 12*E^(2 + x)*x]

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fricas [A]  time = 0.52, size = 33, normalized size = 1.43 \begin {gather*} \log \relax (x) + \log \left (\frac {{\left (7 \, x + 22\right )} e^{8} + 6 \, x e^{\left (2 \, x + 4\right )} + 12 \, x e^{\left (x + 6\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="fricas")

[Out]

log(x) + log(((7*x + 22)*e^8 + 6*x*e^(2*x + 4) + 12*x*e^(x + 6))/x)

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giac [A]  time = 0.26, size = 25, normalized size = 1.09 \begin {gather*} \log \left (7 \, x e^{4} + 6 \, x e^{\left (2 \, x\right )} + 12 \, x e^{\left (x + 2\right )} + 22 \, e^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="giac")

[Out]

log(7*x*e^4 + 6*x*e^(2*x) + 12*x*e^(x + 2) + 22*e^4)

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maple [A]  time = 0.06, size = 28, normalized size = 1.22

method	result	size
risch	\(\ln \relax (x )+\ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{2+x}+\frac {\left (7 x +22\right ) {\mathrm e}^{4}}{6 x}\right )\)	\(28\)
norman	\(\ln \left (7 x \,{\mathrm e}^{4}+12 x \,{\mathrm e}^{2} {\mathrm e}^{x}+6 x \,{\mathrm e}^{2 x}+22 \,{\mathrm e}^{4}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp(2)^2)
,x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(exp(2*x)+2*exp(2+x)+1/6*(7*x+22)*exp(4)/x)

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maxima [A]  time = 0.48, size = 33, normalized size = 1.43 \begin {gather*} \log \relax (x) + \log \left (\frac {7 \, x e^{4} + 6 \, x e^{\left (2 \, x\right )} + 12 \, x e^{\left (x + 2\right )} + 22 \, e^{4}}{6 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)^2+(12*x+12)*exp(2)*exp(x)+7*exp(2)^2)/(6*x*exp(x)^2+12*x*exp(2)*exp(x)+(7*x+22)*exp
(2)^2),x, algorithm="maxima")

[Out]

log(x) + log(1/6*(7*x*e^4 + 6*x*e^(2*x) + 12*x*e^(x + 2) + 22*e^4)/x)

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mupad [B]  time = 2.05, size = 27, normalized size = 1.17 \begin {gather*} \ln \left (22\,{\mathrm {e}}^8+12\,x\,{\mathrm {e}}^{x+6}+7\,x\,{\mathrm {e}}^8+6\,x\,{\mathrm {e}}^{2\,x+4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*exp(4) + exp(2*x)*(12*x + 6) + exp(2)*exp(x)*(12*x + 12))/(6*x*exp(2*x) + exp(4)*(7*x + 22) + 12*x*exp(
2)*exp(x)),x)

[Out]

log(22*exp(8) + 12*x*exp(x + 6) + 7*x*exp(8) + 6*x*exp(2*x + 4))

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sympy [A]  time = 0.29, size = 32, normalized size = 1.39 \begin {gather*} \log {\relax (x )} + \log {\left (e^{2 x} + 2 e^{2} e^{x} + \frac {7 x e^{4} + 22 e^{4}}{6 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x+6)*exp(x)**2+(12*x+12)*exp(2)*exp(x)+7*exp(2)**2)/(6*x*exp(x)**2+12*x*exp(2)*exp(x)+(7*x+22)*
exp(2)**2),x)

[Out]

log(x) + log(exp(2*x) + 2*exp(2)*exp(x) + (7*x*exp(4) + 22*exp(4))/(6*x))

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