3.57.88 \(\int \frac {e^{\frac {x}{4 \log (x)}} (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+(e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7) \log (x)+(4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8)) \log ^2(x))}{4 x^5 \log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ e^{\frac {x}{4 \log (x)}} \left (e^{3+\left (2-\frac {5}{x}+x\right )^4}+x+x^2\right ) \]

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Rubi [B]  time = 4.87, antiderivative size = 146, normalized size of antiderivative = 4.71, number of steps used = 2, number of rules used = 2, integrand size = 236, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {12, 2288} \begin {gather*} \frac {e^{\frac {x}{4 \log (x)}} \left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )-\left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+x^7+x^6\right ) \log (x)+x^7+x^6\right )}{x^5 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x/(4*Log[x]))*(-(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^
5) - x^6 - x^7 + (E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6
+ x^7)*Log[x] + (4*x^5 + 8*x^6 + E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8
)/x^4)*(-10000 + 12000*x - 800*x^2 - 1760*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8))*Log[x]^2))/(4*x^5*Log[x]^
2),x]

[Out]

(E^(x/(4*Log[x]))*(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^
6 + x^7 - (E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6 + x^7)
*Log[x]))/(x^5*(Log[x]^(-2) - Log[x]^(-1))*Log[x]^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

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,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{\frac {x}{4 \log (x)}} \left (-\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5-x^6-x^7+\left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{x^5 \log ^2(x)} \, dx\\ &=\frac {e^{\frac {x}{4 \log (x)}} \left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7-\left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7\right ) \log (x)\right )}{x^5 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 62, normalized size = 2.00 \begin {gather*} e^{-71+\frac {x}{4 \log (x)}} \left (e^{\frac {625-1000 x+100 x^2+440 x^3-88 x^5+4 x^6+8 x^7+x^8}{x^4}}+e^{71} x (1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x/(4*Log[x]))*(-(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x
^4)*x^5) - x^6 - x^7 + (E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5
+ x^6 + x^7)*Log[x] + (4*x^5 + 8*x^6 + E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7
+ x^8)/x^4)*(-10000 + 12000*x - 800*x^2 - 1760*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8))*Log[x]^2))/(4*x^5*L
og[x]^2),x]

[Out]

E^(-71 + x/(4*Log[x]))*(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4) + E^71*x*(1
+ x))

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fricas [B]  time = 0.57, size = 57, normalized size = 1.84 \begin {gather*} {\left (x^{2} + x + e^{\left (\frac {x^{8} + 8 \, x^{7} + 4 \, x^{6} - 88 \, x^{5} - 71 \, x^{4} + 440 \, x^{3} + 100 \, x^{2} - 1000 \, x + 625}{x^{4}}\right )}\right )} e^{\left (\frac {x}{4 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-10000)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x
^4+440*x^3+100*x^2-1000*x+625)/x^4)+8*x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*
x^2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)-x
^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x, algorithm="fricas")

[Out]

(x^2 + x + e^((x^8 + 8*x^7 + 4*x^6 - 88*x^5 - 71*x^4 + 440*x^3 + 100*x^2 - 1000*x + 625)/x^4))*e^(1/4*x/log(x)
)

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giac [B]  time = 0.68, size = 95, normalized size = 3.06 \begin {gather*} x^{2} e^{\left (\frac {x}{4 \, \log \relax (x)}\right )} + x e^{\left (\frac {x}{4 \, \log \relax (x)}\right )} + e^{\left (\frac {4 \, x^{8} \log \relax (x) + 32 \, x^{7} \log \relax (x) + 16 \, x^{6} \log \relax (x) - 352 \, x^{5} \log \relax (x) + x^{5} - 284 \, x^{4} \log \relax (x) + 1760 \, x^{3} \log \relax (x) + 400 \, x^{2} \log \relax (x) - 4000 \, x \log \relax (x) + 2500 \, \log \relax (x)}{4 \, x^{4} \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-10000)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x
^4+440*x^3+100*x^2-1000*x+625)/x^4)+8*x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*
x^2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)-x
^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x, algorithm="giac")

[Out]

x^2*e^(1/4*x/log(x)) + x*e^(1/4*x/log(x)) + e^(1/4*(4*x^8*log(x) + 32*x^7*log(x) + 16*x^6*log(x) - 352*x^5*log
(x) + x^5 - 284*x^4*log(x) + 1760*x^3*log(x) + 400*x^2*log(x) - 4000*x*log(x) + 2500*log(x))/(x^4*log(x)))

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maple [B]  time = 0.10, size = 65, normalized size = 2.10

method	result	size
risch	\(\frac {\left (4 x^{2}+4 x +4 \,{\mathrm e}^{\frac {x^{8}+8 x^{7}+4 x^{6}-88 x^{5}-71 x^{4}+440 x^{3}+100 x^{2}-1000 x +625}{x^{4}}}\right ) {\mathrm e}^{\frac {x}{4 \ln \relax (x )}}}{4}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-10000)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440
*x^3+100*x^2-1000*x+625)/x^4)+8*x^6+4*x^5)*ln(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-100
0*x+625)/x^4)+x^7+x^6)*ln(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)-x^7-x^6)*
exp(1/4*x/ln(x))/x^5/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*(4*x^2+4*x+4*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4))*exp(1/4*x/ln(x))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-10000)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x
^4+440*x^3+100*x^2-1000*x+625)/x^4)+8*x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*
x^2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)-x
^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 4.05, size = 78, normalized size = 2.52 \begin {gather*} x\,{\mathrm {e}}^{\frac {x}{4\,\ln \relax (x)}}+x^2\,{\mathrm {e}}^{\frac {x}{4\,\ln \relax (x)}}+{\mathrm {e}}^{-88\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-71}\,{\mathrm {e}}^{\frac {x}{4\,\ln \relax (x)}}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{8\,x^3}\,{\mathrm {e}}^{\frac {100}{x^2}}\,{\mathrm {e}}^{440/x}\,{\mathrm {e}}^{\frac {625}{x^4}}\,{\mathrm {e}}^{-\frac {1000}{x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x/(4*log(x)))*(x^5*exp((100*x^2 - 1000*x + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8 + 625)/x^
4) - log(x)^2*(exp((100*x^2 - 1000*x + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8 + 625)/x^4)*(12000*x -
800*x^2 - 1760*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8 - 10000) + 4*x^5 + 8*x^6) - log(x)*(x^5*exp((100*x^2 -
1000*x + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8 + 625)/x^4) + x^6 + x^7) + x^6 + x^7))/(4*x^5*log(x)
^2),x)

[Out]

x*exp(x/(4*log(x))) + x^2*exp(x/(4*log(x))) + exp(-88*x)*exp(x^4)*exp(-71)*exp(x/(4*log(x)))*exp(4*x^2)*exp(8*
x^3)*exp(100/x^2)*exp(440/x)*exp(625/x^4)*exp(-1000/x^3)

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sympy [B]  time = 39.86, size = 65, normalized size = 2.10 \begin {gather*} \left (x^{2} + x\right ) e^{\frac {x}{4 \log {\relax (x )}}} + e^{\frac {x^{8} + 8 x^{7} + 4 x^{6} - 88 x^{5} - 71 x^{4} + 440 x^{3} + 100 x^{2} - 1000 x + 625}{x^{4}}} e^{\frac {x}{4 \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((16*x**8+96*x**7+32*x**6-352*x**5-1760*x**3-800*x**2+12000*x-10000)*exp((x**8+8*x**7+4*x**6-88
*x**5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)+8*x**6+4*x**5)*ln(x)**2+(x**5*exp((x**8+8*x**7+4*x**6-88*x**
5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)+x**7+x**6)*ln(x)-x**5*exp((x**8+8*x**7+4*x**6-88*x**5-71*x**4+44
0*x**3+100*x**2-1000*x+625)/x**4)-x**7-x**6)*exp(1/4*x/ln(x))/x**5/ln(x)**2,x)

[Out]

(x**2 + x)*exp(x/(4*log(x))) + exp((x**8 + 8*x**7 + 4*x**6 - 88*x**5 - 71*x**4 + 440*x**3 + 100*x**2 - 1000*x
+ 625)/x**4)*exp(x/(4*log(x)))

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