Integrand size = 94, antiderivative size = 24 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=5 e^{-\left (\frac {5}{x}-9 (-5+x-\log (9))\right )^2} x \] Output:
exp(ln(5*x)-(5/x-9*x+18*ln(3)+45)^2)
Time = 2.92 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=5\ 9^{-18 \left (45+\frac {5}{x}-9 x\right )} e^{-1935-\frac {25}{x^2}-\frac {450}{x}+810 x-81 x^2-81 \log ^2(9)} x \] Input:
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]
Output:
(5*E^(-1935 - 25/x^2 - 450/x + 810*x - 81*x^2 - 81*Log[9]^2)*x)/9^(18*(45 + 5/x - 9*x))
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(24)=48\).
Time = 1.89 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {7257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-162 x^4+810 x^3+\left (162 x^3+90 x\right ) \log (9)+x^2+450 x+50\right ) \exp \left (\frac {-81 x^4+810 x^3-1935 x^2-81 x^2 \log ^2(9)+x^2 \log (5 x)+\left (162 x^3-810 x^2-90 x\right ) \log (9)-450 x-25}{x^2}\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle 5\ 9^{-\frac {-162 x^3+810 x^2+90 x}{x^2}} x \exp \left (-\frac {81 x^4-810 x^3+1935 x^2+81 x^2 \log ^2(9)+450 x+25}{x^2}\right )\) |
Input:
Int[(E^((-25 - 450*x - 1935*x^2 + 810*x^3 - 81*x^4 + (-90*x - 810*x^2 + 16 2*x^3)*Log[9] - 81*x^2*Log[9]^2 + x^2*Log[5*x])/x^2)*(50 + 450*x + x^2 + 8 10*x^3 - 162*x^4 + (90*x + 162*x^3)*Log[9]))/x^3,x]
Output:
(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))
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs. \(2(24)=48\).
Time = 0.85 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54
| method | result | size |
| norman | \({\mathrm e}^{\frac {x^{2} \ln \left (5 x \right )-324 x^{2} \ln \left (3\right )^{2}+2 \left (162 x^{3}-810 x^{2}-90 x \right ) \ln \left (3\right )-81 x^{4}+810 x^{3}-1935 x^{2}-450 x -25}{x^{2}}}\) | \(61\) |
| parallelrisch | \({\mathrm e}^{\frac {x^{2} \ln \left (5 x \right )-324 x^{2} \ln \left (3\right )^{2}+2 \left (162 x^{3}-810 x^{2}-90 x \right ) \ln \left (3\right )-81 x^{4}+810 x^{3}-1935 x^{2}-450 x -25}{x^{2}}}\) | \(61\) |
| gosper | \({\mathrm e}^{-\frac {324 x^{2} \ln \left (3\right )^{2}-324 x^{3} \ln \left (3\right )+81 x^{4}+1620 x^{2} \ln \left (3\right )-x^{2} \ln \left (5 x \right )-810 x^{3}+180 x \ln \left (3\right )+1935 x^{2}+450 x +25}{x^{2}}}\) | \(64\) |
| risch | \({\mathrm e}^{-\frac {324 x^{2} \ln \left (3\right )^{2}-324 x^{3} \ln \left (3\right )+81 x^{4}+1620 x^{2} \ln \left (3\right )-x^{2} \ln \left (5 x \right )-810 x^{3}+180 x \ln \left (3\right )+1935 x^{2}+450 x +25}{x^{2}}}\) | \(64\) |
Input:
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^2-45 0*x-25)/x^2)/x^3,x,method=_RETURNVERBOSE)
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=e^{\left (-\frac {81 \, x^{4} + 324 \, x^{2} \log \left (3\right )^{2} - 810 \, x^{3} - x^{2} \log \left (5 \, x\right ) + 1935 \, x^{2} - 36 \, {\left (9 \, x^{3} - 45 \, x^{2} - 5 \, x\right )} \log \left (3\right ) + 450 \, x + 25}{x^{2}}\right )} \] Input:
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^3-1 935*x^2-450*x-25)/x^2)/x^3,x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=e^{\frac {- 81 x^{4} + 810 x^{3} + x^{2} \log {\left (5 x \right )} - 1935 x^{2} - 324 x^{2} \log {\left (3 \right )}^{2} - 450 x + \left (324 x^{3} - 1620 x^{2} - 180 x\right ) \log {\left (3 \right )} - 25}{x^{2}}} \] Input:
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)
Output:
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 - 180*x)*log(3) - 25)/x**2)
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=\frac {5}{86383867871673826572978861593354619709321595508485014582990287188689462436872226650900170661043350993254637605029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269326610028276279072547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334662263108134935239462783872941285026598053272563292621639606298708727433962291991485280632308262684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063929382671957896693499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401} \, x e^{\left (-81 \, x^{2} + 324 \, x \log \left (3\right ) - 324 \, \log \left (3\right )^{2} + 810 \, x - \frac {180 \, \log \left (3\right )}{x} - \frac {450}{x} - \frac {25}{x^{2}} - 1935\right )} \] Input:
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^3-1 935*x^2-450*x-25)/x^2)/x^3,x, algorithm="maxima")
Output:
5/863838678716738265729788615933546197093215955084850145829902871886894624 36872226650900170661043350993254637605029418092439720229755340868074170726 30129632139009730718878847951913958796418314250032148237718932877588126932 66100282762790725472534612289241046649358010015220709929792349274391878407 07470959385793060329379283707416545532751864980334662263108134935239462783 87294128502659805327256329262163960629870872743396229199148528063230826268 45177548164759451484034728126573817332250024212645818839680020612183858719 56150848906843214599207272427063929382671957896693499203116961768099411781 70464923607376463030729661939081006369987841037062196503263836327386828700 19078480359755771471511056111749781974419838515554855522294118760674867558 45380695890336993315151043395936401*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)
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=e^{\left (-81 \, x^{2} + 324 \, x \log \left (3\right ) - 324 \, \log \left (3\right )^{2} + 810 \, x - \frac {180 \, \log \left (3\right )}{x} - \frac {450}{x} - \frac {25}{x^{2}} - 1620 \, \log \left (3\right ) + \log \left (5 \, x\right ) - 1935\right )} \] Input:
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^3-1 935*x^2-450*x-25)/x^2)/x^3,x, algorithm="giac")
Output:
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) - 1935)
Time = 2.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=\frac {5\,3^{324\,x}\,x\,{\mathrm {e}}^{810\,x}\,{\mathrm {e}}^{-1935}\,{\mathrm {e}}^{-324\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{-\frac {25}{x^2}}\,{\mathrm {e}}^{-81\,x^2}\,{\mathrm {e}}^{-\frac {450}{x}}}{86383867871673826572978861593354619709321595508485014582990287188689462436872226650900170661043350993254637605029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269326610028276279072547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334662263108134935239462783872941285026598053272563292621639606298708727433962291991485280632308262684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063929382671957896693499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401\,3^{180/x}} \] Input:
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)
Output:
(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))/(8638386787167382657297886159335461970932159550848501 45829902871886894624368722266509001706610433509932546376050294180924397202 29755340868074170726301296321390097307188788479519139587964183142500321482 37718932877588126932661002827627907254725346122892410466493580100152207099 29792349274391878407074709593857930603293792837074165455327518649803346622 63108134935239462783872941285026598053272563292621639606298708727433962291 99148528063230826268451775481647594514840347281265738173322500242126458188 39680020612183858719561508489068432145992072724270639293826719578966934992 03116961768099411781704649236073764630307296619390810063699878410370621965 03263836327386828700190784803597557714715110561117497819744198385155548555 2229411876067486755845380695890336993315151043395936401*3^(180/x))
Time = 14.90 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\frac {-25-450 x-1935 x^2+810 x^3-81 x^4+\left (-90 x-810 x^2+162 x^3\right ) \log (9)-81 x^2 \log ^2(9)+x^2 \log (5 x)}{x^2}} \left (50+450 x+x^2+810 x^3-162 x^4+\left (90 x+162 x^3\right ) \log (9)\right )}{x^3} \, dx=\frac {5 e^{810 x} 3^{324 x} x}{86383867871673826572978861593354619709321595508485014582990287188689462436872226650900170661043350993254637605029418092439720229755340868074170726301296321390097307188788479519139587964183142500321482377189328775881269326610028276279072547253461228924104664935801001522070992979234927439187840707470959385793060329379283707416545532751864980334662263108134935239462783872941285026598053272563292621639606298708727433962291991485280632308262684517754816475945148403472812657381733225002421264581883968002061218385871956150848906843214599207272427063929382671957896693499203116961768099411781704649236073764630307296619390810063699878410370621965032638363273868287001907848035975577147151105611174978197441983851555485552229411876067486755845380695890336993315151043395936401 e^{\frac {324 x^{2} \mathrm {log}\left (3\right )^{2}+81 x^{4}+450 x +25}{x^{2}}} 3^{\frac {180}{x}} e^{1935}} \] Input:
int((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)
Output:
(5*e**(810*x)*3**(324*x)*x)/(863838678716738265729788615933546197093215955 08485014582990287188689462436872226650900170661043350993254637605029418092 43972022975534086807417072630129632139009730718878847951913958796418314250 03214823771893287758812693266100282762790725472534612289241046649358010015 22070992979234927439187840707470959385793060329379283707416545532751864980 33466226310813493523946278387294128502659805327256329262163960629870872743 39622919914852806323082626845177548164759451484034728126573817332250024212 64581883968002061218385871956150848906843214599207272427063929382671957896 69349920311696176809941178170464923607376463030729661939081006369987841037 06219650326383632738682870019078480359755771471511056111749781974419838515 55485552229411876067486755845380695890336993315151043395936401*e**((324*lo g(3)**2*x**2 + 81*x**4 + 450*x + 25)/x**2)*3**(180/x)*e**1935)