∫ ex(−140+8x−4x2+5 log(3)+16 log(5))_______________________ 19600+1120x2+16x4+(−1400−40x2) log(3)+25 log2(3)+(−4480−128x2+160 log(3)) log(5)+256 log2(5) dx [8306]

Optimal. Leaf size=25 \[ \frac {e^x}{8+5 \log (3)-4 \left (1+x^2-4 (-9+\log (5))\right )} \]

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(25)=50\).
time = 0.28, antiderivative size = 73, normalized size of antiderivative = 2.92, number of steps used = 1, number of rules used = 1, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2326} \begin {gather*} -\frac {e^x \left (4 x^2+140-\log (243)-16 \log (5)\right )}{16 x^4+1120 x^2-32 \log (5) \left (4 x^2+5 (28-\log (3))\right )-40 \left (x^2+35\right ) \log (3)+19600+256 \log ^2(5)+25 \log ^2(3)} \end {gather*}

Int[(E^x*(-140 + 8*x - 4*x^2 + 5*Log[3] + 16*Log[5]))/(19600 + 1120*x^2 + 16*x^4 + (-1400 - 40*x^2)*Log[3] + 2

5*Log[3]^2 + (-4480 - 128*x^2 + 160*Log[3])*Log[5] + 256*Log[5]^2),x]

-((E^x*(140 + 4*x^2 - 16*Log[5] - Log[243]))/(19600 + 1120*x^2 + 16*x^4 - 40*(35 + x^2)*Log[3] + 25*Log[3]^2 -

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {e^x \left (140+4 x^2-16 \log (5)-\log (243)\right )}{19600+1120 x^2+16 x^4-40 \left (35+x^2\right ) \log (3)+25 \log ^2(3)-32 \left (4 x^2+5 (28-\log (3))\right ) \log (5)+256 \log ^2(5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.20, size = 34, normalized size = 1.36 \begin {gather*} \frac {e^x \left (-140-4 x^2+16 \log (5)+\log (243)\right )}{\left (140+4 x^2-5 \log (3)-16 \log (5)\right )^2} \end {gather*}

Integrate[(E^x*(-140 + 8*x - 4*x^2 + 5*Log[3] + 16*Log[5]))/(19600 + 1120*x^2 + 16*x^4 + (-1400 - 40*x^2)*Log[

3] + 25*Log[3]^2 + (-4480 - 128*x^2 + 160*Log[3])*Log[5] + 256*Log[5]^2),x]

(E^x*(-140 - 4*x^2 + 16*Log[5] + Log[243]))/(140 + 4*x^2 - 5*Log[3] - 16*Log[5])^2

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.52, size = 1038, normalized size = 41.52


method	result	size

gosper	\(\frac {{\mathrm e}^{x}}{-140-4 x^{2}+16 \ln \left (5\right )+5 \ln \left (3\right )}\)	\(21\)
norman	\(\frac {{\mathrm e}^{x}}{-140-4 x^{2}+16 \ln \left (5\right )+5 \ln \left (3\right )}\)	\(21\)
default	\(\text {Expression too large to display}\)	\(1038\)

int((16*ln(5)+5*ln(3)-4*x^2+8*x-140)*exp(x)/(256*ln(5)^2+(160*ln(3)-128*x^2-4480)*ln(5)+25*ln(3)^2+(-40*x^2-14

00)*ln(3)+16*x^4+1120*x^2+19600),x,method=_RETURNVERBOSE)

-70*exp(x)*x/(16*ln(5)+5*ln(3)-140)/(-140-4*x^2+16*ln(5)+5*ln(3))+35/4/(16*ln(5)+5*ln(3)-140)*exp(1/2*I*(-16*l

n(5)-5*ln(3)+140)^(1/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+35/4/(16*ln(5)+5*ln(3)-140)*exp(-1/2*I*(

-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+2*I*ln(5)/(16*ln(5)+5*ln(3)-140)/(-

16*ln(5)-5*ln(3)+140)^(1/2)*exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1

/2))-1/8*I/(-16*ln(5)-5*ln(3)+140)^(1/2)*exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*

ln(3)+140)^(1/2))+exp(x)/(-140-4*x^2+16*ln(5)+5*ln(3))+35/2*I/(16*ln(5)+5*ln(3)-140)/(-16*ln(5)-5*ln(3)+140)^(

1/2)*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))-5/8*I*ln(3)/(16*ln(

5)+5*ln(3)-140)/(-16*ln(5)-5*ln(3)+140)^(1/2)*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x+1/2*I*(-16*ln(5

)-5*ln(3)+140)^(1/2))-1/2*exp(x)*x/(-140-4*x^2+16*ln(5)+5*ln(3))+1/16*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))

*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+1/16*exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-1

6*ln(5)-5*ln(3)+140)^(1/2))+5/2*ln(3)*exp(x)*x/(16*ln(5)+5*ln(3)-140)/(-140-4*x^2+16*ln(5)+5*ln(3))-5/16*ln(3)

/(16*ln(5)+5*ln(3)-140)*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))-

5/16*ln(3)/(16*ln(5)+5*ln(3)-140)*exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+1

40)^(1/2))-2*I*ln(5)/(16*ln(5)+5*ln(3)-140)/(-16*ln(5)-5*ln(3)+140)^(1/2)*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^(1

/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))-35/2*I/(16*ln(5)+5*ln(3)-140)/(-16*ln(5)-5*ln(3)+140)^(1/2)*

exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+8*ln(5)*exp(x)*x/(16*ln

(5)+5*ln(3)-140)/(-140-4*x^2+16*ln(5)+5*ln(3))-ln(5)/(16*ln(5)+5*ln(3)-140)*exp(1/2*I*(-16*ln(5)-5*ln(3)+140)^

(1/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))-ln(5)/(16*ln(5)+5*ln(3)-140)*exp(-1/2*I*(-16*ln(5)-5*ln(3)

+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+1/8*I/(-16*ln(5)-5*ln(3)+140)^(1/2)*exp(1/2*I*(-16*l

n(5)-5*ln(3)+140)^(1/2))*Ei(1,-x+1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))+5/8*I*ln(3)/(16*ln(5)+5*ln(3)-140)/(-16*

ln(5)-5*ln(3)+140)^(1/2)*exp(-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2))*Ei(1,-x-1/2*I*(-16*ln(5)-5*ln(3)+140)^(1/2)

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Maxima [A]
time = 0.49, size = 21, normalized size = 0.84 \begin {gather*} -\frac {e^{x}}{4 \, x^{2} - 16 \, \log \left (5\right ) - 5 \, \log \left (3\right ) + 140} \end {gather*}

integrate((16*log(5)+5*log(3)-4*x^2+8*x-140)*exp(x)/(256*log(5)^2+(160*log(3)-128*x^2-4480)*log(5)+25*log(3)^2

+(-40*x^2-1400)*log(3)+16*x^4+1120*x^2+19600),x, algorithm="maxima")

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Fricas [A]
time = 0.39, size = 21, normalized size = 0.84 \begin {gather*} -\frac {e^{x}}{4 \, x^{2} - 16 \, \log \left (5\right ) - 5 \, \log \left (3\right ) + 140} \end {gather*}

integrate((16*log(5)+5*log(3)-4*x^2+8*x-140)*exp(x)/(256*log(5)^2+(160*log(3)-128*x^2-4480)*log(5)+25*log(3)^2

+(-40*x^2-1400)*log(3)+16*x^4+1120*x^2+19600),x, algorithm="fricas")

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Sympy [A]
time = 0.08, size = 20, normalized size = 0.80 \begin {gather*} - \frac {e^{x}}{4 x^{2} - 16 \log {\left (5 \right )} - 5 \log {\left (3 \right )} + 140} \end {gather*}

integrate((16*ln(5)+5*ln(3)-4*x**2+8*x-140)*exp(x)/(256*ln(5)**2+(160*ln(3)-128*x**2-4480)*ln(5)+25*ln(3)**2+(

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Giac [A]
time = 0.41, size = 21, normalized size = 0.84 \begin {gather*} -\frac {e^{x}}{4 \, x^{2} - 16 \, \log \left (5\right ) - 5 \, \log \left (3\right ) + 140} \end {gather*}

integrate((16*log(5)+5*log(3)-4*x^2+8*x-140)*exp(x)/(256*log(5)^2+(160*log(3)-128*x^2-4480)*log(5)+25*log(3)^2

+(-40*x^2-1400)*log(3)+16*x^4+1120*x^2+19600),x, algorithm="giac")

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (-4\,x^2+8\,x+5\,\ln \left (3\right )+16\,\ln \left (5\right )-140\right )}{25\,{\ln \left (3\right )}^2-\ln \left (3\right )\,\left (40\,x^2+1400\right )-\ln \left (5\right )\,\left (128\,x^2-160\,\ln \left (3\right )+4480\right )+256\,{\ln \left (5\right )}^2+1120\,x^2+16\,x^4+19600} \,d x \end {gather*}

int((exp(x)*(8*x + 5*log(3) + 16*log(5) - 4*x^2 - 140))/(25*log(3)^2 - log(3)*(40*x^2 + 1400) - log(5)*(128*x^

2 - 160*log(3) + 4480) + 256*log(5)^2 + 1120*x^2 + 16*x^4 + 19600),x)

int((exp(x)*(8*x + 5*log(3) + 16*log(5) - 4*x^2 - 140))/(25*log(3)^2 - log(3)*(40*x^2 + 1400) - log(5)*(128*x^

2 - 160*log(3) + 4480) + 256*log(5)^2 + 1120*x^2 + 16*x^4 + 19600), x)

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3.84.6 \(\int \frac {e^x (-140+8 x-4 x^2+5 \log (3)+16 \log (5))}{19600+1120 x^2+16 x^4+(-1400-40 x^2) \log (3)+25 \log ^2(3)+(-4480-128 x^2+160 \log (3)) \log (5)+256 \log ^2(5)} \, dx\) [8306]