3.32.14 \(\int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+(-90 x-810 x^2+162 x^3) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} (50+450 x+x^2+810 x^3-162 x^4+(90 x+162 x^3) \log (9))}{x^3} \, dx\)

Optimal. Leaf size=24 \[ 5 e^{-\left (\frac {5}{x}-9 (-5+x-\log (9))\right )^2} x \]

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Rubi [B]  time = 2.44, antiderivative size = 60, normalized size of antiderivative = 2.50, number of steps used = 1, number of rules used = 1, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} 5\ 9^{-\frac {-162 x^3+810 x^2+90 x}{x^2}} x \exp \left (-\frac {81 x^4-810 x^3+1935 x^2+81 x^2 \log ^2(9)+450 x+25}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-25 - 450*x - 1935*x^2 + 810*x^3 - 81*x^4 + (-90*x - 810*x^2 + 162*x^3)*Log[9] - 81*x^2*Log[9]^2 + x^
2*Log[5*x])/x^2)*(50 + 450*x + x^2 + 810*x^3 - 162*x^4 + (90*x + 162*x^3)*Log[9]))/x^3,x]

[Out]

(5*x)/(9^((90*x + 810*x^2 - 162*x^3)/x^2)*E^((25 + 450*x + 1935*x^2 - 810*x^3 + 81*x^4 + 81*x^2*Log[9]^2)/x^2)
)

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} &=5\ 9^{-\frac {90 x+810 x^2-162 x^3}{x^2}} \exp \left (-\frac {25+450 x+1935 x^2-810 x^3+81 x^4+81 x^2 \log ^2(9)}{x^2}\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.10, size = 43, normalized size = 1.79 \begin {gather*} 5\ 9^{-810-\frac {90}{x}+162 x} e^{-1935-\frac {25}{x^2}-\frac {450}{x}+810 x-81 x^2-81 \log ^2(9)} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-25 - 450*x - 1935*x^2 + 810*x^3 - 81*x^4 + (-90*x - 810*x^2 + 162*x^3)*Log[9] - 81*x^2*Log[9]^
2 + x^2*Log[5*x])/x^2)*(50 + 450*x + x^2 + 810*x^3 - 162*x^4 + (90*x + 162*x^3)*Log[9]))/x^3,x]

[Out]

5*9^(-810 - 90/x + 162*x)*E^(-1935 - 25/x^2 - 450/x + 810*x - 81*x^2 - 81*Log[9]^2)*x

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fricas [B]  time = 0.64, size = 62, normalized size = 2.58 \begin {gather*} e^{\left (-\frac {81 \, x^{4} + 324 \, x^{2} \log \relax (3)^{2} - 810 \, x^{3} - x^{2} \log \left (5 \, x\right ) + 1935 \, x^{2} - 36 \, {\left (9 \, x^{3} - 45 \, x^{2} - 5 \, x\right )} \log \relax (3) + 450 \, x + 25}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(162*x^3+90*x)*log(3)-162*x^4+810*x^3+x^2+450*x+50)*exp((x^2*log(5*x)-324*x^2*log(3)^2+2*(162*x^3
-810*x^2-90*x)*log(3)-81*x^4+810*x^3-1935*x^2-450*x-25)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(-(81*x^4 + 324*x^2*log(3)^2 - 810*x^3 - x^2*log(5*x) + 1935*x^2 - 36*(9*x^3 - 45*x^2 - 5*x)*log(3) + 450*x
+ 25)/x^2)

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giac [A]  time = 0.21, size = 47, normalized size = 1.96 \begin {gather*} e^{\left (-81 \, x^{2} + 324 \, x \log \relax (3) - 324 \, \log \relax (3)^{2} + 810 \, x - \frac {180 \, \log \relax (3)}{x} - \frac {450}{x} - \frac {25}{x^{2}} - 1620 \, \log \relax (3) + \log \left (5 \, x\right ) - 1935\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(162*x^3+90*x)*log(3)-162*x^4+810*x^3+x^2+450*x+50)*exp((x^2*log(5*x)-324*x^2*log(3)^2+2*(162*x^3
-810*x^2-90*x)*log(3)-81*x^4+810*x^3-1935*x^2-450*x-25)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(-81*x^2 + 324*x*log(3) - 324*log(3)^2 + 810*x - 180*log(3)/x - 450/x - 25/x^2 - 1620*log(3) + log(5*x) - 19
35)

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maple [B]  time = 0.08, size = 61, normalized size = 2.54

method	result	size
norman	\({\mathrm e}^{\frac {x^{2} \ln \left (5 x \right )-324 x^{2} \ln \relax (3)^{2}+2 \left (162 x^{3}-810 x^{2}-90 x \right ) \ln \relax (3)-81 x^{4}+810 x^{3}-1935 x^{2}-450 x -25}{x^{2}}}\)	\(61\)
gosper	\({\mathrm e}^{\frac {-324 x^{2} \ln \relax (3)^{2}+324 x^{3} \ln \relax (3)-81 x^{4}+x^{2} \ln \left (5 x \right )-1620 x^{2} \ln \relax (3)+810 x^{3}-180 x \ln \relax (3)-1935 x^{2}-450 x -25}{x^{2}}}\)	\(62\)
risch	\({\mathrm e}^{-\frac {324 x^{2} \ln \relax (3)^{2}-324 x^{3} \ln \relax (3)+81 x^{4}-x^{2} \ln \left (5 x \right )+1620 x^{2} \ln \relax (3)-810 x^{3}+180 x \ln \relax (3)+1935 x^{2}+450 x +25}{x^{2}}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(162*x^3+90*x)*ln(3)-162*x^4+810*x^3+x^2+450*x+50)*exp((x^2*ln(5*x)-324*x^2*ln(3)^2+2*(162*x^3-810*x^2-
90*x)*ln(3)-81*x^4+810*x^3-1935*x^2-450*x-25)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp((x^2*ln(5*x)-324*x^2*ln(3)^2+2*(162*x^3-810*x^2-90*x)*ln(3)-81*x^4+810*x^3-1935*x^2-450*x-25)/x^2)

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maxima [A]  time = 0.88, size = 42, normalized size = 1.75 \begin {gather*} \frac {5}{86383867871673826572978861593354619709321595508485014582990287188689462436872226650900170661043350993254637605029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269326610028276279072547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334662263108134935239462783872941285026598053272563292621639606298708727433962291991485280632308262684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063929382671957896693499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401} \, x e^{\left (-81 \, x^{2} + 324 \, x \log \relax (3) - 324 \, \log \relax (3)^{2} + 810 \, x - \frac {180 \, \log \relax (3)}{x} - \frac {450}{x} - \frac {25}{x^{2}} - 1935\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(162*x^3+90*x)*log(3)-162*x^4+810*x^3+x^2+450*x+50)*exp((x^2*log(5*x)-324*x^2*log(3)^2+2*(162*x^3
-810*x^2-90*x)*log(3)-81*x^4+810*x^3-1935*x^2-450*x-25)/x^2)/x^3,x, algorithm="maxima")

[Out]

5/863838678716738265729788615933546197093215955084850145829902871886894624368722266509001706610433509932546376
05029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269
32661002827627907254725346122892410466493580100152207099297923492743918784070747095938579306032937928370741654
55327518649803346622631081349352394627838729412850265980532725632926216396062987087274339622919914852806323082
62684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063
92938267195789669349920311696176809941178170464923607376463030729661939081006369987841037062196503263836327386
82870019078480359755771471511056111749781974419838515554855522294118760674867558453806958903369933151510433959
36401*x*e^(-81*x^2 + 324*x*log(3) - 324*log(3)^2 + 810*x - 180*log(3)/x - 450/x - 25/x^2 - 1935)

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mupad [B]  time = 2.07, size = 48, normalized size = 2.00 \begin {gather*} \frac {5\,3^{324\,x}\,x\,{\mathrm {e}}^{810\,x}\,{\mathrm {e}}^{-1935}\,{\mathrm {e}}^{-324\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-\frac {25}{x^2}}\,{\mathrm {e}}^{-81\,x^2}\,{\mathrm {e}}^{-\frac {450}{x}}}{86383867871673826572978861593354619709321595508485014582990287188689462436872226650900170661043350993254637605029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269326610028276279072547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334662263108134935239462783872941285026598053272563292621639606298708727433962291991485280632308262684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063929382671957896693499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401\,3^{180/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(450*x + 324*x^2*log(3)^2 + 2*log(3)*(90*x + 810*x^2 - 162*x^3) - x^2*log(5*x) + 1935*x^2 - 810*x^3
+ 81*x^4 + 25)/x^2)*(450*x + 2*log(3)*(90*x + 162*x^3) + x^2 + 810*x^3 - 162*x^4 + 50))/x^3,x)

[Out]

(5*3^(324*x)*x*exp(810*x)*exp(-1935)*exp(-324*log(3)^2)*exp(-25/x^2)*exp(-81*x^2)*exp(-450/x))/(86383867871673
82657297886159335461970932159550848501458299028718868946243687222665090017066104335099325463760502941809243972
02297553408680741707263012963213900973071887884795191395879641831425003214823771893287758812693266100282762790
72547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334
66226310813493523946278387294128502659805327256329262163960629870872743396229199148528063230826268451775481647
59451484034728126573817332250024212645818839680020612183858719561508489068432145992072724270639293826719578966
93499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035
975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401*3^(180/x))

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sympy [B]  time = 0.53, size = 60, normalized size = 2.50 \begin {gather*} e^{\frac {- 81 x^{4} + 810 x^{3} + x^{2} \log {\left (5 x \right )} - 1935 x^{2} - 324 x^{2} \log {\relax (3 )}^{2} - 450 x + \left (324 x^{3} - 1620 x^{2} - 180 x\right ) \log {\relax (3 )} - 25}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(162*x**3+90*x)*ln(3)-162*x**4+810*x**3+x**2+450*x+50)*exp((x**2*ln(5*x)-324*x**2*ln(3)**2+2*(162
*x**3-810*x**2-90*x)*ln(3)-81*x**4+810*x**3-1935*x**2-450*x-25)/x**2)/x**3,x)

[Out]

exp((-81*x**4 + 810*x**3 + x**2*log(5*x) - 1935*x**2 - 324*x**2*log(3)**2 - 450*x + (324*x**3 - 1620*x**2 - 18
0*x)*log(3) - 25)/x**2)

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