3.21.39 \(\int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8)}{(4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8) \log (2)} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^{-\frac {5}{-2-x+x^2 (4+x)^2}} x}{\log (2)} \]

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Rubi [F]  time = 5.70, 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 {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8\right ) \log (2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - x + 97*x^2 + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4
))*(4 + 4*x - 63*x^2 - 64*x^3 + 236*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)*Log[2]),x]

[Out]

Defer[Int][E^(-5/(-2 - x + 16*x^2 + 8*x^3 + x^4)), x]/Log[2] + (40*Defer[Int][1/(E^(5/(-2 - x + 16*x^2 + 8*x^3
+ x^4))*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] + (15*Defer[Int][x/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^
4))*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] - (160*Defer[Int][x^2/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4)
)*(-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] - (40*Defer[Int][x^3/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(
-2 - x + 16*x^2 + 8*x^3 + x^4)^2), x])/Log[2] + (20*Defer[Int][1/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*(-2 -
x + 16*x^2 + 8*x^3 + x^4)), x])/Log[2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{4+4 x-63 x^2-64 x^3+236 x^4+254 x^5+96 x^6+16 x^7+x^8} \, dx}{\log (2)}\\ &=\frac {\int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (4-x+97 x^2+56 x^3+256 x^4+254 x^5+96 x^6+16 x^7+x^8\right )}{\left (2+x-16 x^2-8 x^3-x^4\right )^2} \, dx}{\log (2)}\\ &=\frac {\int \left (e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}-\frac {5 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (-8-3 x+32 x^2+8 x^3\right )}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {20 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4}\right ) \, dx}{\log (2)}\\ &=\frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}-\frac {5 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \left (-8-3 x+32 x^2+8 x^3\right )}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)}\\ &=\frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}-\frac {5 \int \left (-\frac {8 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}-\frac {3 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {32 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^2}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}+\frac {8 e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^3}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2}\right ) \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)}\\ &=\frac {\int e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} \, dx}{\log (2)}+\frac {15 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}+\frac {20 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{-2-x+16 x^2+8 x^3+x^4} \, dx}{\log (2)}+\frac {40 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}}}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}-\frac {40 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^3}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}-\frac {160 \int \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x^2}{\left (-2-x+16 x^2+8 x^3+x^4\right )^2} \, dx}{\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 30, normalized size = 1.15 \begin {gather*} \frac {e^{-\frac {5}{-2-x+16 x^2+8 x^3+x^4}} x}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - x + 97*x^2 + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)/(E^(5/(-2 - x + 16*x^2 + 8*x^3
+ x^4))*(4 + 4*x - 63*x^2 - 64*x^3 + 236*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8)*Log[2]),x]

[Out]

x/(E^(5/(-2 - x + 16*x^2 + 8*x^3 + x^4))*Log[2])

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fricas [A]  time = 0.58, size = 29, normalized size = 1.12 \begin {gather*} \frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x^6+254*x^5+236*x^4-64*x^3-63*x
^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x^2-x-2)),x, algorithm="fricas")

[Out]

x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)

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giac [A]  time = 0.27, size = 48, normalized size = 1.85 \begin {gather*} \frac {x e^{\left (-\frac {5 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x\right )}}{2 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2\right )}} + \frac {5}{2}\right )}}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x^6+254*x^5+236*x^4-64*x^3-63*x
^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x^2-x-2)),x, algorithm="giac")

[Out]

x*e^(-5/2*(x^4 + 8*x^3 + 16*x^2 - x)/(x^4 + 8*x^3 + 16*x^2 - x - 2) + 5/2)/log(2)

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maple [A]  time = 0.16, size = 30, normalized size = 1.15

method	result	size
risch	\(\frac {x \,{\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{\ln \relax (2)}\)	\(30\)
gosper	\(\frac {x \,{\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{\ln \relax (2)}\)	\(32\)
norman	\(\frac {\left (\frac {x^{5}}{\ln \relax (2)}-\frac {2 x}{\ln \relax (2)}-\frac {x^{2}}{\ln \relax (2)}+\frac {16 x^{3}}{\ln \relax (2)}+\frac {8 x^{4}}{\ln \relax (2)}\right ) {\mathrm e}^{-\frac {5}{x^{4}+8 x^{3}+16 x^{2}-x -2}}}{x^{4}+8 x^{3}+16 x^{2}-x -2}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x^6+254*x^5+236*x^4-64*x^3-63*x^2+4*x
+4)/ln(2)/exp(5/(x^4+8*x^3+16*x^2-x-2)),x,method=_RETURNVERBOSE)

[Out]

x*exp(-5/(x^4+8*x^3+16*x^2-x-2))/ln(2)

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maxima [A]  time = 0.52, size = 29, normalized size = 1.12 \begin {gather*} \frac {x e^{\left (-\frac {5}{x^{4} + 8 \, x^{3} + 16 \, x^{2} - x - 2}\right )}}{\log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+16*x^7+96*x^6+254*x^5+256*x^4+56*x^3+97*x^2-x+4)/(x^8+16*x^7+96*x^6+254*x^5+236*x^4-64*x^3-63*x
^2+4*x+4)/log(2)/exp(5/(x^4+8*x^3+16*x^2-x-2)),x, algorithm="maxima")

[Out]

x*e^(-5/(x^4 + 8*x^3 + 16*x^2 - x - 2))/log(2)

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mupad [B]  time = 1.61, size = 29, normalized size = 1.12 \begin {gather*} \frac {x\,{\mathrm {e}}^{-\frac {5}{x^4+8\,x^3+16\,x^2-x-2}}}{\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5/(16*x^2 - x + 8*x^3 + x^4 - 2))*(97*x^2 - x + 56*x^3 + 256*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8 +
4))/(log(2)*(4*x - 63*x^2 - 64*x^3 + 236*x^4 + 254*x^5 + 96*x^6 + 16*x^7 + x^8 + 4)),x)

[Out]

(x*exp(-5/(16*x^2 - x + 8*x^3 + x^4 - 2)))/log(2)

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sympy [A]  time = 1.31, size = 24, normalized size = 0.92 \begin {gather*} \frac {x e^{- \frac {5}{x^{4} + 8 x^{3} + 16 x^{2} - x - 2}}}{\log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**8+16*x**7+96*x**6+254*x**5+256*x**4+56*x**3+97*x**2-x+4)/(x**8+16*x**7+96*x**6+254*x**5+236*x**4
-64*x**3-63*x**2+4*x+4)/ln(2)/exp(5/(x**4+8*x**3+16*x**2-x-2)),x)

[Out]

x*exp(-5/(x**4 + 8*x**3 + 16*x**2 - x - 2))/log(2)

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