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

Optimal. Leaf size=24 \[ x \left (4 e^{\frac {x}{-3+3 (-254-2 x)^2+x}}+x\right ) \]

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Rubi [F]  time = 0.60, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

x^2 + 4*Defer[Int][E^(x/(193545 + 3049*x + 12*x^2)), x] + (611524*Defer[Int][E^(x/(193545 + 3049*x + 12*x^2))/
(391 + 3*x)^2, x])/79 - (1564*Defer[Int][E^(x/(193545 + 3049*x + 12*x^2))/(391 + 3*x), x])/79 - (980100*Defer[
Int][E^(x/(193545 + 3049*x + 12*x^2))/(495 + 4*x)^2, x])/79 + (1980*Defer[Int][E^(x/(193545 + 3049*x + 12*x^2)
)/(495 + 4*x), x])/79

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+\frac {4 e^{\frac {x}{193545+3049 x+12 x^2}} \left (37459667025+1180430955 x+13941481 x^2+73164 x^3+144 x^4\right )}{\left (193545+3049 x+12 x^2\right )^2}\right ) \, dx\\ &=x^2+4 \int \frac {e^{\frac {x}{193545+3049 x+12 x^2}} \left (37459667025+1180430955 x+13941481 x^2+73164 x^3+144 x^4\right )}{\left (193545+3049 x+12 x^2\right )^2} \, dx\\ &=x^2+4 \int \left (e^{\frac {x}{193545+3049 x+12 x^2}}+\frac {152881 e^{\frac {x}{193545+3049 x+12 x^2}}}{79 (391+3 x)^2}-\frac {391 e^{\frac {x}{193545+3049 x+12 x^2}}}{79 (391+3 x)}-\frac {245025 e^{\frac {x}{193545+3049 x+12 x^2}}}{79 (495+4 x)^2}+\frac {495 e^{\frac {x}{193545+3049 x+12 x^2}}}{79 (495+4 x)}\right ) \, dx\\ &=x^2+4 \int e^{\frac {x}{193545+3049 x+12 x^2}} \, dx-\frac {1564}{79} \int \frac {e^{\frac {x}{193545+3049 x+12 x^2}}}{391+3 x} \, dx+\frac {1980}{79} \int \frac {e^{\frac {x}{193545+3049 x+12 x^2}}}{495+4 x} \, dx+\frac {611524}{79} \int \frac {e^{\frac {x}{193545+3049 x+12 x^2}}}{(391+3 x)^2} \, dx-\frac {980100}{79} \int \frac {e^{\frac {x}{193545+3049 x+12 x^2}}}{(495+4 x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 23, normalized size = 0.96 \begin {gather*} 4 e^{\frac {x}{193545+3049 x+12 x^2}} x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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 + 139414
81*x^2 + 73176*x^3 + 144*x^4),x]

[Out]

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

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fricas [A]  time = 1.18, size = 22, normalized size = 0.92 \begin {gather*} x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp(x/(12*x^2+3049*x+193545))+288*x^5+1
46352*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")

[Out]

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

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giac [A]  time = 0.21, size = 22, normalized size = 0.92 \begin {gather*} x^{2} + 4 \, x e^{\left (\frac {x}{12 \, x^{2} + 3049 \, x + 193545}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp(x/(12*x^2+3049*x+193545))+288*x^5+1
46352*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")

[Out]

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

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maple [A]  time = 0.31, size = 25, normalized size = 1.04

method	result	size
risch	\(x^{2}+4 \,{\mathrm e}^{\frac {x}{\left (3 x +391\right ) \left (4 x +495\right )}} x\)	\(25\)
norman	\(\frac {-\frac {196706235 x}{4}+3049 x^{3}+12 x^{4}+774180 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x +12196 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{2}+48 \,{\mathrm e}^{\frac {x}{12 x^{2}+3049 x +193545}} x^{3}-\frac {12486555675}{4}}{12 x^{2}+3049 x +193545}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+74919334050*x)/(144*x^4+73176*x^3+13941481*x^2+1180237410*x+37459667025),x,met
hod=_RETURNVERBOSE)

[Out]

x^2+4*exp(x/(3*x+391)/(4*x+495))*x

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maxima [B]  time = 0.50, size = 122, normalized size = 5.08 \begin {gather*} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((576*x^4+292656*x^3+55765924*x^2+4721723820*x+149838668100)*exp(x/(12*x^2+3049*x+193545))+288*x^5+1
46352*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")

[Out]

x^2 + 4*x*e^(-495/79/(4*x + 495) + 391/79/(3*x + 391)) + 1/449352*(16579595834289949*x + 2099267057378616345)/
(12*x^2 + 3049*x + 193545) - 3049/224676*(10846402941841*x + 1374257227948605)/(12*x^2 + 3049*x + 193545) + 13
941481/449352*(7100453269*x + 900239922945)/(12*x^2 + 3049*x + 193545) - 196706235/6241*(4651321*x + 590118705
)/(12*x^2 + 3049*x + 193545) + 74919334050/6241*(3049*x + 387090)/(12*x^2 + 3049*x + 193545)

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mupad [B]  time = 0.35, size = 22, normalized size = 0.92 \begin {gather*} x^2+4\,x\,{\mathrm {e}}^{\frac {x}{12\,x^2+3049\,x+193545}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((74919334050*x + exp(x/(3049*x + 12*x^2 + 193545))*(4721723820*x + 55765924*x^2 + 292656*x^3 + 576*x^4 + 1
49838668100) + 2360474820*x^2 + 27882962*x^3 + 146352*x^4 + 288*x^5)/(1180237410*x + 13941481*x^2 + 73176*x^3
+ 144*x^4 + 37459667025),x)

[Out]

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

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sympy [A]  time = 0.27, size = 19, normalized size = 0.79 \begin {gather*} x^{2} + 4 x e^{\frac {x}{12 x^{2} + 3049 x + 193545}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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+3
7459667025),x)

[Out]

x**2 + 4*x*exp(x/(12*x**2 + 3049*x + 193545))

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