3.12.22 \(\int \frac {e^{\frac {49+1134 x^7+6561 e^x x^{12}+6561 x^{13}+6561 x^{14}}{6561 x^{12}}} (-196-1890 x^7+2187 x^{13}+2187 e^x x^{13}+4374 x^{14})}{2187 x^{13}} \, dx\)

Optimal. Leaf size=18 \[ e^{e^x+x+\left (\frac {7}{81 x^6}+x\right )^2} \]

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Rubi [A]  time = 1.36, antiderivative size = 34, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 2, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {12, 6706} \begin {gather*} e^{\frac {6561 x^{14}+6561 x^{13}+6561 e^x x^{12}+1134 x^7+49}{6561 x^{12}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((49 + 1134*x^7 + 6561*E^x*x^12 + 6561*x^13 + 6561*x^14)/(6561*x^12))*(-196 - 1890*x^7 + 2187*x^13 + 21
87*E^x*x^13 + 4374*x^14))/(2187*x^13),x]

[Out]

E^((49 + 1134*x^7 + 6561*E^x*x^12 + 6561*x^13 + 6561*x^14)/(6561*x^12))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {e^{\frac {49+1134 x^7+6561 e^x x^{12}+6561 x^{13}+6561 x^{14}}{6561 x^{12}}} \left (-196-1890 x^7+2187 x^{13}+2187 e^x x^{13}+4374 x^{14}\right )}{x^{13}} \, dx}{2187}\\ &=e^{\frac {49+1134 x^7+6561 e^x x^{12}+6561 x^{13}+6561 x^{14}}{6561 x^{12}}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 24, normalized size = 1.33 \begin {gather*} e^{e^x+\frac {49}{6561 x^{12}}+\frac {14}{81 x^5}+x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((49 + 1134*x^7 + 6561*E^x*x^12 + 6561*x^13 + 6561*x^14)/(6561*x^12))*(-196 - 1890*x^7 + 2187*x^1
3 + 2187*E^x*x^13 + 4374*x^14))/(2187*x^13),x]

[Out]

E^(E^x + 49/(6561*x^12) + 14/(81*x^5) + x + x^2)

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fricas [A]  time = 0.71, size = 30, normalized size = 1.67 \begin {gather*} e^{\left (\frac {6561 \, x^{14} + 6561 \, x^{13} + 6561 \, x^{12} e^{x} + 1134 \, x^{7} + 49}{6561 \, x^{12}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2187*(2187*x^13*exp(x)+4374*x^14+2187*x^13-1890*x^7-196)*exp(1/6561*(6561*x^12*exp(x)+6561*x^14+65
61*x^13+1134*x^7+49)/x^12)/x^13,x, algorithm="fricas")

[Out]

e^(1/6561*(6561*x^14 + 6561*x^13 + 6561*x^12*e^x + 1134*x^7 + 49)/x^12)

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giac [A]  time = 0.33, size = 18, normalized size = 1.00 \begin {gather*} e^{\left (x^{2} + x + \frac {14}{81 \, x^{5}} + \frac {49}{6561 \, x^{12}} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2187*(2187*x^13*exp(x)+4374*x^14+2187*x^13-1890*x^7-196)*exp(1/6561*(6561*x^12*exp(x)+6561*x^14+65
61*x^13+1134*x^7+49)/x^12)/x^13,x, algorithm="giac")

[Out]

e^(x^2 + x + 14/81/x^5 + 49/6561/x^12 + e^x)

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maple [B]  time = 0.05, size = 31, normalized size = 1.72

method	result	size
risch	\({\mathrm e}^{\frac {6561 x^{12} {\mathrm e}^{x}+6561 x^{14}+6561 x^{13}+1134 x^{7}+49}{6561 x^{12}}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2187*(2187*x^13*exp(x)+4374*x^14+2187*x^13-1890*x^7-196)*exp(1/6561*(6561*x^12*exp(x)+6561*x^14+6561*x^1
3+1134*x^7+49)/x^12)/x^13,x,method=_RETURNVERBOSE)

[Out]

exp(1/6561*(6561*x^12*exp(x)+6561*x^14+6561*x^13+1134*x^7+49)/x^12)

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maxima [A]  time = 0.75, size = 18, normalized size = 1.00 \begin {gather*} e^{\left (x^{2} + x + \frac {14}{81 \, x^{5}} + \frac {49}{6561 \, x^{12}} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2187*(2187*x^13*exp(x)+4374*x^14+2187*x^13-1890*x^7-196)*exp(1/6561*(6561*x^12*exp(x)+6561*x^14+65
61*x^13+1134*x^7+49)/x^12)/x^13,x, algorithm="maxima")

[Out]

e^(x^2 + x + 14/81/x^5 + 49/6561/x^12 + e^x)

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mupad [B]  time = 0.93, size = 22, normalized size = 1.22 \begin {gather*} {\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {14}{81\,x^5}}\,{\mathrm {e}}^{\frac {49}{6561\,x^{12}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^12*exp(x) + (14*x^7)/81 + x^13 + x^14 + 49/6561)/x^12)*(2187*x^13*exp(x) - 1890*x^7 + 2187*x^13 +
4374*x^14 - 196))/(2187*x^13),x)

[Out]

exp(x^2)*exp(exp(x))*exp(14/(81*x^5))*exp(49/(6561*x^12))*exp(x)

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sympy [A]  time = 0.28, size = 27, normalized size = 1.50 \begin {gather*} e^{\frac {x^{14} + x^{13} + x^{12} e^{x} + \frac {14 x^{7}}{81} + \frac {49}{6561}}{x^{12}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2187*(2187*x**13*exp(x)+4374*x**14+2187*x**13-1890*x**7-196)*exp(1/6561*(6561*x**12*exp(x)+6561*x*
*14+6561*x**13+1134*x**7+49)/x**12)/x**13,x)

[Out]

exp((x**14 + x**13 + x**12*exp(x) + 14*x**7/81 + 49/6561)/x**12)

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