3.8.56 \(\int \frac {e^{1+8 x+x^2+x \log (\frac {7-3 e^4+6 x}{3 x})} (-49-62 x-12 x^2+e^4 (21+6 x)+(-7+3 e^4-6 x) \log (\frac {7-3 e^4+6 x}{3 x}))}{-7+3 e^4-6 x} \, dx\)

Optimal. Leaf size=25 \[ e^{1+x \left (8+x+\log \left (2+\frac {\frac {7}{3}-e^4}{x}\right )\right )} \]

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Rubi [A]  time = 0.55, antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 1, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} 3^{-x} e^{x^2+8 x+1} \left (\frac {6 x-3 e^4+7}{x}\right )^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(1 + 8*x + x^2 + x*Log[(7 - 3*E^4 + 6*x)/(3*x)])*(-49 - 62*x - 12*x^2 + E^4*(21 + 6*x) + (-7 + 3*E^4 -
6*x)*Log[(7 - 3*E^4 + 6*x)/(3*x)]))/(-7 + 3*E^4 - 6*x),x]

[Out]

(E^(1 + 8*x + x^2)*((7 - 3*E^4 + 6*x)/x)^x)/3^x

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3^{-x} e^{1+8 x+x^2} \left (\frac {7-3 e^4+6 x}{x}\right )^x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 32, normalized size = 1.28 \begin {gather*} 3^{-x} e^{1+8 x+x^2} \left (\frac {7-3 e^4+6 x}{x}\right )^x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 + 8*x + x^2 + x*Log[(7 - 3*E^4 + 6*x)/(3*x)])*(-49 - 62*x - 12*x^2 + E^4*(21 + 6*x) + (-7 + 3*
E^4 - 6*x)*Log[(7 - 3*E^4 + 6*x)/(3*x)]))/(-7 + 3*E^4 - 6*x),x]

[Out]

(E^(1 + 8*x + x^2)*((7 - 3*E^4 + 6*x)/x)^x)/3^x

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fricas [A]  time = 0.59, size = 26, normalized size = 1.04 \begin {gather*} e^{\left (x^{2} + x \log \left (\frac {6 \, x - 3 \, e^{4} + 7}{3 \, x}\right ) + 8 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(4)-6*x-7)*log(1/3*(-3*exp(4)+6*x+7)/x)+(6*x+21)*exp(4)-12*x^2-62*x-49)*exp(x*log(1/3*(-3*exp
(4)+6*x+7)/x)+x^2+8*x+1)/(3*exp(4)-6*x-7),x, algorithm="fricas")

[Out]

e^(x^2 + x*log(1/3*(6*x - 3*e^4 + 7)/x) + 8*x + 1)

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giac [A]  time = 0.92, size = 26, normalized size = 1.04 \begin {gather*} e^{\left (x^{2} + x \log \left (-\frac {e^{4}}{x} + \frac {7}{3 \, x} + 2\right ) + 8 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(4)-6*x-7)*log(1/3*(-3*exp(4)+6*x+7)/x)+(6*x+21)*exp(4)-12*x^2-62*x-49)*exp(x*log(1/3*(-3*exp
(4)+6*x+7)/x)+x^2+8*x+1)/(3*exp(4)-6*x-7),x, algorithm="giac")

[Out]

e^(x^2 + x*log(-e^4/x + 7/3/x + 2) + 8*x + 1)

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maple [A]  time = 0.20, size = 27, normalized size = 1.08

method	result	size
norman	\({\mathrm e}^{x \ln \left (\frac {-3 \,{\mathrm e}^{4}+6 x +7}{3 x}\right )+x^{2}+8 x +1}\)	\(27\)
risch	\(\left (\frac {-3 \,{\mathrm e}^{4}+6 x +7}{3 x}\right )^{x} {\mathrm e}^{x^{2}+8 x +1}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*exp(4)-6*x-7)*ln(1/3*(-3*exp(4)+6*x+7)/x)+(6*x+21)*exp(4)-12*x^2-62*x-49)*exp(x*ln(1/3*(-3*exp(4)+6*x+
7)/x)+x^2+8*x+1)/(3*exp(4)-6*x-7),x,method=_RETURNVERBOSE)

[Out]

exp(x*ln(1/3*(-3*exp(4)+6*x+7)/x)+x^2+8*x+1)

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maxima [A]  time = 1.00, size = 31, normalized size = 1.24 \begin {gather*} e^{\left (x^{2} - x \log \relax (3) + x \log \left (6 \, x - 3 \, e^{4} + 7\right ) - x \log \relax (x) + 8 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(4)-6*x-7)*log(1/3*(-3*exp(4)+6*x+7)/x)+(6*x+21)*exp(4)-12*x^2-62*x-49)*exp(x*log(1/3*(-3*exp
(4)+6*x+7)/x)+x^2+8*x+1)/(3*exp(4)-6*x-7),x, algorithm="maxima")

[Out]

e^(x^2 - x*log(3) + x*log(6*x - 3*e^4 + 7) - x*log(x) + 8*x + 1)

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mupad [B]  time = 1.02, size = 26, normalized size = 1.04 \begin {gather*} {\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{x^2}\,\mathrm {e}\,{\left (\frac {2\,x-{\mathrm {e}}^4+\frac {7}{3}}{x}\right )}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8*x + x*log((2*x - exp(4) + 7/3)/x) + x^2 + 1)*(62*x + 12*x^2 + log((2*x - exp(4) + 7/3)/x)*(6*x - 3*
exp(4) + 7) - exp(4)*(6*x + 21) + 49))/(6*x - 3*exp(4) + 7),x)

[Out]

exp(8*x)*exp(x^2)*exp(1)*((2*x - exp(4) + 7/3)/x)^x

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sympy [A]  time = 0.58, size = 24, normalized size = 0.96 \begin {gather*} e^{x^{2} + x \log {\left (\frac {2 x - e^{4} + \frac {7}{3}}{x} \right )} + 8 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*exp(4)-6*x-7)*ln(1/3*(-3*exp(4)+6*x+7)/x)+(6*x+21)*exp(4)-12*x**2-62*x-49)*exp(x*ln(1/3*(-3*exp(
4)+6*x+7)/x)+x**2+8*x+1)/(3*exp(4)-6*x-7),x)

[Out]

exp(x**2 + x*log((2*x - exp(4) + 7/3)/x) + 8*x + 1)

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