Integrand size = 84, antiderivative size = 24 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x \left (4 e^{\frac {x}{-3+3 (-254-2 x)^2+x}}+x\right ) \] Output:
x*(x+4*exp(x/(x+3*(-254-2*x)^2-3)))
Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=4 e^{\frac {x}{193545+3049 x+12 x^2}} x+x^2 \] Input:
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]
Output:
4*E^(x/(193545 + 3049*x + 12*x^2))*x + x^2
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 |
\(\displaystyle 4 \int e^{\frac {x}{12 x^2+3049 x+193545}}dx+\frac {611524}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{(3 x+391)^2}dx-\frac {1564}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{3 x+391}dx-\frac {980100}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{(4 x+495)^2}dx+\frac {1980}{79} \int \frac {e^{\frac {x}{12 x^2+3049 x+193545}}}{4 x+495}dx+x^2\) |
Input:
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]
Output:
$Aborted
Time = 1.06 (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 | \(\frac {-\frac {196706235 x}{4}+3049 x^{3}+12 x^{4}+774180 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x +12196 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{2}+48 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{3}-\frac {12486555675}{4}}{12 x^{2}+3049 x +193545}\) | \(87\) |
orering | \(\frac {\left (1990656 x^{9}+2023170048 x^{8}+899549998848 x^{7}+228524848285824 x^{6}+36283604220488592 x^{5}+3687418219466183528 x^{4}+234247217663629356120 x^{3}+8496229553874401684400 x^{2}+133648038248909870597175 x -67385044753384820145750\right ) \left (\left (576 x^{4}+292656 x^{3}+55765924 x^{2}+4721723820 x +149838668100\right ) {\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}}+288 x^{5}+146352 x^{4}+27882962 x^{3}+2360474820 x^{2}+74919334050 x \right )}{96 \left (20736 x^{8}+21072960 x^{7}+9369873360 x^{6}+2380522910604 x^{5}+377940038045641 x^{4}+38394858894404265 x^{3}+2437446778694569125 x^{2}+88415350646842456875 x +1403226653623872350625\right ) \left (144 x^{4}+73176 x^{3}+13941481 x^{2}+1180237410 x +37459667025\right )}-\frac {\left (3 x +391\right )^{2} \left (4 x +495\right )^{2} \left (20736 x^{6}+10539072 x^{5}-850088199324 x^{3}-188970700421761 x^{2}-16456955730944355 x -522243236095364025\right ) \left (\frac {\left (2304 x^{3}+877968 x^{2}+111531848 x +4721723820\right ) {\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}}+\left (576 x^{4}+292656 x^{3}+55765924 x^{2}+4721723820 x +149838668100\right ) \left (\frac {1}{12 x^{2}+3049 x +193545}-\frac {x \left (24 x +3049\right )}{\left (12 x^{2}+3049 x +193545\right )^{2}}\right ) {\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}}+1440 x^{4}+585408 x^{3}+83648886 x^{2}+4720949640 x +74919334050}{144 x^{4}+73176 x^{3}+13941481 x^{2}+1180237410 x +37459667025}-\frac {\left (\left (576 x^{4}+292656 x^{3}+55765924 x^{2}+4721723820 x +149838668100\right ) {\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}}+288 x^{5}+146352 x^{4}+27882962 x^{3}+2360474820 x^{2}+74919334050 x \right ) \left (576 x^{3}+219528 x^{2}+27882962 x +1180237410\right )}{\left (144 x^{4}+73176 x^{3}+13941481 x^{2}+1180237410 x +37459667025\right )^{2}}\right )}{288 \left (20736 x^{8}+21072960 x^{7}+9369873360 x^{6}+2380522910604 x^{5}+377940038045641 x^{4}+38394858894404265 x^{3}+2437446778694569125 x^{2}+88415350646842456875 x +1403226653623872350625\right )}\) | \(499\) |
Input:
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)
Output:
x^2+4*exp(x/(12*x^2+3049*x+193545))*x-10844761/24
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \] Input:
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="fricas")
Output:
x^2 + 4*x*e^(x/(12*x^2 + 3049*x + 193545))
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 x e^{\frac {x}{12 x^{2} + 3049 x + 193545}} \] Input:
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)
Output:
x**2 + 4*x*exp(x/(12*x**2 + 3049*x + 193545))
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (21) = 42\).
Time = 0.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 5.08 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (-\frac {495}{79 \, {\left (4 \, x + 495\right )}} + \frac {391}{79 \, {\left (3 \, x + 391\right )}}\right )} + \frac {16579595834289949 \, x + 2099267057378616345}{449352 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} - \frac {3049 \, {\left (10846402941841 \, x + 1374257227948605\right )}}{224676 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} + \frac {13941481 \, {\left (7100453269 \, x + 900239922945\right )}}{449352 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} - \frac {196706235 \, {\left (4651321 \, x + 590118705\right )}}{6241 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} + \frac {74919334050 \, {\left (3049 \, x + 387090\right )}}{6241 \, {\left (12 \, x^{2} + 3049 \, x + 193545\right )}} \] Input:
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="maxima")
Output:
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 = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \] Input:
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="giac")
Output:
x^2 + 4*x*e^(x/(12*x^2 + 3049*x + 193545))
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x^2+4\,x\,{\mathrm {e}}^{\frac {x}{12\,x^2+3049\,x+193545}} \] Input:
int((74919334050*x + exp(x/(3049*x + 12*x^2 + 193545))*(4721723820*x + 557 65924*x^2 + 292656*x^3 + 576*x^4 + 149838668100) + 2360474820*x^2 + 278829 62*x^3 + 146352*x^4 + 288*x^5)/(1180237410*x + 13941481*x^2 + 73176*x^3 + 144*x^4 + 37459667025),x)
Output:
x^2 + 4*x*exp(x/(3049*x + 12*x^2 + 193545))
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} \left (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4\right )}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx=x \left (4 e^{\frac {x}{12 x^{2}+3049 x +193545}}+x \right ) \] Input:
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)
Output:
x*(4*e**(x/(12*x**2 + 3049*x + 193545)) + x)