Integrand size = 84, antiderivative size = 24 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x \left (4 e^{\frac {x}{-3+3 (-254-2 x)^2+x}}+x\right ) \end {dmath*}
Time = 0.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=4 e^{\frac {x}{193545+3049 x+12 x^2}} x+x^2 \end {dmath*}
Integrate[(74919334050*x + 2360474820*x^2 + 27882962*x^3 + 146352*x^4 + 28 8*x^5 + E^(x/(193545 + 3049*x + 12*x^2))*(149838668100 + 4721723820*x + 55 765924*x^2 + 292656*x^3 + 576*x^4))/(37459667025 + 1180237410*x + 13941481 *x^2 + 73176*x^3 + 144*x^4),x]
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.
\(\displaystyle \int \frac {288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )+74919334050 x}{144 x^4+73176 x^3+13941481 x^2+1180237410 x+37459667025} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {72 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )+74919334050 x\right )}{493039 (3 x+391)}-\frac {96 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )+74919334050 x\right )}{493039 (4 x+495)}+\frac {9 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )+74919334050 x\right )}{6241 (3 x+391)^2}+\frac {16 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )+74919334050 x\right )}{6241 (4 x+495)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int e^{\frac {x}{12 x^2+3049 x+193545}}dx+\frac {611524}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{(3 x+391)^2}dx-\frac {1564}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{3 x+391}dx-\frac {980100}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{(4 x+495)^2}dx+\frac {1980}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{4 x+495}dx+x^2\) |
Int[(74919334050*x + 2360474820*x^2 + 27882962*x^3 + 146352*x^4 + 288*x^5 + E^(x/(193545 + 3049*x + 12*x^2))*(149838668100 + 4721723820*x + 55765924 *x^2 + 292656*x^3 + 576*x^4))/(37459667025 + 1180237410*x + 13941481*x^2 + 73176*x^3 + 144*x^4),x]
3.5.5.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(x^{2}+4 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x -\frac {10844761}{24}\) | \(24\) |
risch | \(x^{2}+4 \,{\mathrm e}^{\frac {x}{\left (3 x +391\right ) \left (4 x +495\right )}} x\) | \(25\) |
parts | \(x^{2}+\frac {774180 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x +12196 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{2}+48 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{3}}{12 x^{2}+3049 x +193545}\) | \(77\) |
norman | \(\frac {-\frac {196706235 x}{4}+3049 x^{3}+12 x^{4}+774180 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x +12196 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{2}+48 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{3}-\frac {12486555675}{4}}{12 x^{2}+3049 x +193545}\) | \(87\) |
int(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp(x/(12 *x^2+3049*x+193545))+288*x^5+146352*x^4+27882962*x^3+2360474820*x^2+749193 34050*x)/(144*x^4+73176*x^3+13941481*x^2+1180237410*x+37459667025),x,metho d=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \end {dmath*}
integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp (x/(12*x^2+3049*x+193545))+288*x^5+146352*x^4+27882962*x^3+2360474820*x^2+ 74919334050*x)/(144*x^4+73176*x^3+13941481*x^2+1180237410*x+37459667025),x , algorithm=\
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 x e^{\frac {x}{12 x^{2} + 3049 x + 193545}} \end {dmath*}
integrate(((576*x**4+292656*x**3+55765924*x**2+4721723820*x+149838668100)* exp(x/(12*x**2+3049*x+193545))+288*x**5+146352*x**4+27882962*x**3+23604748 20*x**2+74919334050*x)/(144*x**4+73176*x**3+13941481*x**2+1180237410*x+374 59667025),x)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (21) = 42\).
Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 5.08 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (-\frac {495}{79 \, {\left (4 \, x + 495\right )}} + \frac {391}{79 \, {\left (3 \, x + 391\right )}}\right )} + \frac {16579595834289949 \, x + 2099267057378616345}{449352 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} - \frac {3049 \, {\left (10846402941841 \, x + 1374257227948605\right )}}{224676 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} + \frac {13941481 \, {\left (7100453269 \, x + 900239922945\right )}}{449352 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} - \frac {196706235 \, {\left (4651321 \, x + 590118705\right )}}{6241 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} + \frac {74919334050 \, {\left (3049 \, x + 387090\right )}}{6241 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} \end {dmath*}
integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp (x/(12*x^2+3049*x+193545))+288*x^5+146352*x^4+27882962*x^3+2360474820*x^2+ 74919334050*x)/(144*x^4+73176*x^3+13941481*x^2+1180237410*x+37459667025),x , algorithm=\
x^2 + 4*x*e^(-495/79/(4*x + 495) + 391/79/(3*x + 391)) + 1/449352*(1657959 5834289949*x + 2099267057378616345)/(12*x^2 + 3049*x + 193545) - 3049/2246 76*(10846402941841*x + 1374257227948605)/(12*x^2 + 3049*x + 193545) + 1394 1481/449352*(7100453269*x + 900239922945)/(12*x^2 + 3049*x + 193545) - 196 706235/6241*(4651321*x + 590118705)/(12*x^2 + 3049*x + 193545) + 749193340 50/6241*(3049*x + 387090)/(12*x^2 + 3049*x + 193545)
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \end {dmath*}
integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp (x/(12*x^2+3049*x+193545))+288*x^5+146352*x^4+27882962*x^3+2360474820*x^2+ 74919334050*x)/(144*x^4+73176*x^3+13941481*x^2+1180237410*x+37459667025),x , algorithm=\
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^2+4\,x\,{\mathrm {e}}^{\frac {x}{12\,x^2+3049\,x+193545}} \end {dmath*}