3.5.5 \(\int \frac {74919334050 x+2360474820 x^2+27882962 x^3+146352 x^4+288 x^5+e^{\frac {x}{193545+3049 x+12 x^2}} (149838668100+4721723820 x+55765924 x^2+292656 x^3+576 x^4)}{37459667025+1180237410 x+13941481 x^2+73176 x^3+144 x^4} \, dx\) [405]

3.5.5.1 Optimal result

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*}

output

x*(x+4*exp(x/(x+3*(-254-2*x)^2-3)))
 

3.5.5.2 Mathematica [A] (verified)

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*}

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
 

3.5.5.3 Rubi [F]

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.

\(\displaystyle \int \frac {288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )[phone] x}{144 x^4+73176 x^3+13941481 x^2+1180237410 x+37459667025} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {72 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )[phone] x\right )}{493039 (3 x+391)}-\frac {96 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )[phone] x\right )}{493039 (4 x+495)}+\frac {9 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )[phone] x\right )}{6241 (3 x+391)^2}+\frac {16 \left (288 x^5+146352 x^4+27882962 x^3+2360474820 x^2+e^{\frac {x}{12 x^2+3049 x+193545}} \left (576 x^4+292656 x^3+55765924 x^2+4721723820 x+149838668100\right )[phone] x\right )}{6241 (4 x+495)^2}\right )dx\)

\(\Big \downarrow \) 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
 

3.5.5.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

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]
 

3.5.5.4 Maple [A] (verified)

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	\(\frac {-\frac {196706235 x}{4}+3049 x^{3}+12 x^{4}+774180 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x +12196 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{2}+48 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{3}-\frac {12486555675}{4}}{12 x^{2}+3049 x +193545}\)	\(87\)

input

int(((576*x^4+292656*x^3+55765924*x^2+4721723820*x[phone])*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[phone]),x,metho 
d=_RETURNVERBOSE)
 

output

x^2+4*exp(x/(12*x^2+3049*x+193545))*x-10844761/24
 

3.5.5.5 Fricas [A] (verification not implemented)

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*}

input

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x[phone])*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[phone]),x 
, algorithm=\
 

output

x^2 + 4*x*e^(x/(12*x^2 + 3049*x + 193545))
 

3.5.5.6 Sympy [A] (verification not implemented)

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*}

input

integrate(((576*x**4+292656*x**3+55765924*x**2+4721723820*x[phone])* 
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))
 

3.5.5.7 Maxima [B] (verification not implemented)

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*}

input

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x[phone])*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[phone]),x 
, algorithm=\
 

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)
 

3.5.5.8 Giac [A] (verification not implemented)

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*}

input

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x[phone])*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[phone]),x 
, algorithm=\
 

output

x^2 + 4*x*e^(x/(12*x^2 + 3049*x + 193545))
 

3.5.5.9 Mupad [B] (verification not implemented)

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*}

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))