3.13.28 \(\int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} (e^{16} (600-114 x^2)+228 e^{16} x \log (2)-114 e^{16} \log ^2(2))}{10000+1000 x+3825 x^2+190 x^3+361 x^4+(-1000-3850 x-380 x^2-722 x^3) \log (2)+(25+190 x+361 x^2) \log ^2(2)} \, dx\)

Optimal. Leaf size=30 \[ 3 e^{\frac {2 e^{16}}{5-x+20 \left (x+\frac {5}{x-\log (2)}\right )}} \]

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Rubi [F]  time = 6.89, 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 {\exp \left (\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}\right ) \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x)*Log[2]))*(E^16*(600 - 114*x^2) + 228*E^1
6*x*Log[2] - 114*E^16*Log[2]^2))/(10000 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*x^2 -
722*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[2]^2),x]

[Out]

(228*(5 + 19*Log[2])*(5 - 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2])*Defer[Int][E^(16 + (-2*E^16*x
+ 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(5 + 38*x - 19*Log[2] + I*Sqrt[7575 - 190*Log[
2] - 361*Log[2]^2])^2, x])/(7575 - 190*Log[2] - 361*Log[2]^2) - (8664*(200 - Log[32] - Log[2]*Log[524288])*Def
er[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(5 + 38*x - 19*Log[
2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2])^2, x])/(7575 - 190*Log[2] - 361*Log[2]^2) - ((228*I)*(5 - 19*Lo
g[2])*(5 + 19*Log[2])*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[
32]))/(5 + 38*x - 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2]), x])/(7575 - 190*Log[2] - 361*Log[2]^2
)^(3/2) - ((228*I)*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]
))/(5 + 38*x - 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2]), x])/Sqrt[7575 - 190*Log[2] - 361*Log[2]^
2] + ((8664*I)*(200 - Log[2]*(5 + Log[524288]))*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2
- x*(5 - 19*Log[2]) + Log[32]))/(5 + 38*x - 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2]), x])/(7575 -
190*Log[2] - 361*Log[2]^2)^(3/2) + (228*(5 + 19*Log[2])*(5 - 19*Log[2] - I*Sqrt[7575 - 190*Log[2] - 361*Log[2
]^2])*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(-5 - 38*x
+ 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2])^2, x])/(7575 - 190*Log[2] - 361*Log[2]^2) - (8664*(20
0 - Log[32] - Log[2]*Log[524288])*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Lo
g[2]) + Log[32]))/(-5 - 38*x + 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2])^2, x])/(7575 - 190*Log[2]
- 361*Log[2]^2) - ((228*I)*(5 - 19*Log[2])*(5 + 19*Log[2])*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-1
00 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(-5 - 38*x + 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2])
, x])/(7575 - 190*Log[2] - 361*Log[2]^2)^(3/2) - ((228*I)*Defer[Int][E^(16 + (-2*E^16*x + 2*E^16*Log[2])/(-100
- 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(-5 - 38*x + 19*Log[2] + I*Sqrt[7575 - 190*Log[2] - 361*Log[2]^2]),
x])/Sqrt[7575 - 190*Log[2] - 361*Log[2]^2] + ((8664*I)*(200 - Log[2]*(5 + Log[524288]))*Defer[Int][E^(16 + (-2
*E^16*x + 2*E^16*Log[2])/(-100 - 19*x^2 - x*(5 - 19*Log[2]) + Log[32]))/(-5 - 38*x + 19*Log[2] + I*Sqrt[7575 -
190*Log[2] - 361*Log[2]^2]), x])/(7575 - 190*Log[2] - 361*Log[2]^2)^(3/2)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) \left (-114 e^{16} x^2+228 e^{16} x \log (2)+6 e^{16} \left (100-19 \log ^2(2)\right )\right )}{361 x^4+38 x^3 (5-19 \log (2))+10 x (5-19 \log (2)) (20-\log (2))+25 (20-\log (2))^2+x^2 \left (3825-380 \log (2)+361 \log ^2(2)\right )} \, dx\\ &=\int \left (\frac {6 \exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{-19 x^2-x (5-19 \log (2))-5 (20-\log (2))}+\frac {6 \exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) \left (200-5 \log (2)-19 \log ^2(2)+x (5+19 \log (2))\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2}\right ) \, dx\\ &=6 \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{-19 x^2-x (5-19 \log (2))-5 (20-\log (2))} \, dx+6 \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) \left (200-5 \log (2)-19 \log ^2(2)+x (5+19 \log (2))\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2} \, dx\\ &=6 \int \left (-\frac {38 i \exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{\sqrt {7575-190 \log (2)-361 \log ^2(2)} \left (5+38 x-19 \log (2)+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}\right )}-\frac {38 i \exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{\sqrt {7575-190 \log (2)-361 \log ^2(2)} \left (-5-38 x+19 \log (2)+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}\right )}\right ) \, dx+6 \int \left (\frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) x (5+19 \log (2))}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2}+\frac {200 \exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) \left (1-\frac {1}{200} \log (2) (5+\log (524288))\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2}\right ) \, dx\\ &=(6 (5+19 \log (2))) \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right ) x}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2} \, dx-\frac {(228 i) \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{5+38 x-19 \log (2)+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}} \, dx}{\sqrt {7575-190 \log (2)-361 \log ^2(2)}}-\frac {(228 i) \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{-5-38 x+19 \log (2)+i \sqrt {7575-190 \log (2)-361 \log ^2(2)}} \, dx}{\sqrt {7575-190 \log (2)-361 \log ^2(2)}}+(6 (200-\log (2) (5+\log (524288)))) \int \frac {\exp \left (16+\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-19 x^2-x (5-19 \log (2))+\log (32)}\right )}{\left (19 x^2+x (5-19 \log (2))+5 (20-\log (2))\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {-2 e^{16} x+2 e^{16} \log (2)}{-100-5 x-19 x^2+(5+19 x) \log (2)}} \left (e^{16} \left (600-114 x^2\right )+228 e^{16} x \log (2)-114 e^{16} \log ^2(2)\right )}{10000+1000 x+3825 x^2+190 x^3+361 x^4+\left (-1000-3850 x-380 x^2-722 x^3\right ) \log (2)+\left (25+190 x+361 x^2\right ) \log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x)*Log[2]))*(E^16*(600 - 114*x^2) + 2
28*E^16*x*Log[2] - 114*E^16*Log[2]^2))/(10000 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*
x^2 - 722*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[2]^2),x]

[Out]

Integrate[(E^((-2*E^16*x + 2*E^16*Log[2])/(-100 - 5*x - 19*x^2 + (5 + 19*x)*Log[2]))*(E^16*(600 - 114*x^2) + 2
28*E^16*x*Log[2] - 114*E^16*Log[2]^2))/(10000 + 1000*x + 3825*x^2 + 190*x^3 + 361*x^4 + (-1000 - 3850*x - 380*
x^2 - 722*x^3)*Log[2] + (25 + 190*x + 361*x^2)*Log[2]^2), x]

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fricas [A]  time = 1.05, size = 37, normalized size = 1.23 \begin {gather*} 3 \, e^{\left (\frac {2 \, {\left (x e^{16} - e^{16} \log \relax (2)\right )}}{19 \, x^{2} - {\left (19 \, x + 5\right )} \log \relax (2) + 5 \, x + 100}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(16))*exp((2*exp(16)*log(2)-2*x*exp(16
))/((19*x+5)*log(2)-19*x^2-5*x-100))/((361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^
4+190*x^3+3825*x^2+1000*x+10000),x, algorithm="fricas")

[Out]

3*e^(2*(x*e^16 - e^16*log(2))/(19*x^2 - (19*x + 5)*log(2) + 5*x + 100))

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giac [B]  time = 0.34, size = 57, normalized size = 1.90 \begin {gather*} 3 \, e^{\left (\frac {2 \, x e^{16}}{19 \, x^{2} - 19 \, x \log \relax (2) + 5 \, x - 5 \, \log \relax (2) + 100} - \frac {2 \, e^{16} \log \relax (2)}{19 \, x^{2} - 19 \, x \log \relax (2) + 5 \, x - 5 \, \log \relax (2) + 100}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(16))*exp((2*exp(16)*log(2)-2*x*exp(16
))/((19*x+5)*log(2)-19*x^2-5*x-100))/((361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^
4+190*x^3+3825*x^2+1000*x+10000),x, algorithm="giac")

[Out]

3*e^(2*x*e^16/(19*x^2 - 19*x*log(2) + 5*x - 5*log(2) + 100) - 2*e^16*log(2)/(19*x^2 - 19*x*log(2) + 5*x - 5*lo
g(2) + 100))

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maple [A]  time = 0.22, size = 35, normalized size = 1.17

method	result	size
gosper	\(3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \left (\ln \relax (2)-x \right )}{19 x \ln \relax (2)-19 x^{2}+5 \ln \relax (2)-5 x -100}}\)	\(35\)
risch	\(3 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \left (\ln \relax (2)-x \right )}{19 x \ln \relax (2)-19 x^{2}+5 \ln \relax (2)-5 x -100}}\)	\(35\)
norman	\(\frac {\left (15 \ln \relax (2)-300\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \relax (2)-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \relax (2)-19 x^{2}-5 x -100}}+\left (57 \ln \relax (2)-15\right ) x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \relax (2)-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \relax (2)-19 x^{2}-5 x -100}}-57 x^{2} {\mathrm e}^{\frac {2 \,{\mathrm e}^{16} \ln \relax (2)-2 x \,{\mathrm e}^{16}}{\left (19 x +5\right ) \ln \relax (2)-19 x^{2}-5 x -100}}}{19 x \ln \relax (2)-19 x^{2}+5 \ln \relax (2)-5 x -100}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-114*exp(16)*ln(2)^2+228*x*exp(16)*ln(2)+(-114*x^2+600)*exp(16))*exp((2*exp(16)*ln(2)-2*x*exp(16))/((19*x
+5)*ln(2)-19*x^2-5*x-100))/((361*x^2+190*x+25)*ln(2)^2+(-722*x^3-380*x^2-3850*x-1000)*ln(2)+361*x^4+190*x^3+38
25*x^2+1000*x+10000),x,method=_RETURNVERBOSE)

[Out]

3*exp(2*exp(16)*(ln(2)-x)/(19*x*ln(2)-19*x^2+5*ln(2)-5*x-100))

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maxima [B]  time = 0.78, size = 59, normalized size = 1.97 \begin {gather*} 3 \, e^{\left (\frac {2 \, x e^{16}}{19 \, x^{2} - x {\left (19 \, \log \relax (2) - 5\right )} - 5 \, \log \relax (2) + 100} - \frac {2 \, e^{16} \log \relax (2)}{19 \, x^{2} - x {\left (19 \, \log \relax (2) - 5\right )} - 5 \, \log \relax (2) + 100}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114*exp(16)*log(2)^2+228*x*exp(16)*log(2)+(-114*x^2+600)*exp(16))*exp((2*exp(16)*log(2)-2*x*exp(16
))/((19*x+5)*log(2)-19*x^2-5*x-100))/((361*x^2+190*x+25)*log(2)^2+(-722*x^3-380*x^2-3850*x-1000)*log(2)+361*x^
4+190*x^3+3825*x^2+1000*x+10000),x, algorithm="maxima")

[Out]

3*e^(2*x*e^16/(19*x^2 - x*(19*log(2) - 5) - 5*log(2) + 100) - 2*e^16*log(2)/(19*x^2 - x*(19*log(2) - 5) - 5*lo
g(2) + 100))

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mupad [B]  time = 3.55, size = 58, normalized size = 1.93 \begin {gather*} \frac {3\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{16}}{5\,x-5\,\ln \relax (2)-19\,x\,\ln \relax (2)+19\,x^2+100}}}{2^{\frac {2\,{\mathrm {e}}^{16}}{5\,x-5\,\ln \relax (2)-19\,x\,\ln \relax (2)+19\,x^2+100}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*exp(16)*log(2) - 2*x*exp(16))/(5*x - log(2)*(19*x + 5) + 19*x^2 + 100))*(exp(16)*(114*x^2 - 600)
+ 114*exp(16)*log(2)^2 - 228*x*exp(16)*log(2)))/(1000*x - log(2)*(3850*x + 380*x^2 + 722*x^3 + 1000) + log(2)
^2*(190*x + 361*x^2 + 25) + 3825*x^2 + 190*x^3 + 361*x^4 + 10000),x)

[Out]

(3*exp((2*x*exp(16))/(5*x - 5*log(2) - 19*x*log(2) + 19*x^2 + 100)))/2^((2*exp(16))/(5*x - 5*log(2) - 19*x*log
(2) + 19*x^2 + 100))

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sympy [A]  time = 0.97, size = 36, normalized size = 1.20 \begin {gather*} 3 e^{\frac {- 2 x e^{16} + 2 e^{16} \log {\relax (2 )}}{- 19 x^{2} - 5 x + \left (19 x + 5\right ) \log {\relax (2 )} - 100}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-114*exp(16)*ln(2)**2+228*x*exp(16)*ln(2)+(-114*x**2+600)*exp(16))*exp((2*exp(16)*ln(2)-2*x*exp(16)
)/((19*x+5)*ln(2)-19*x**2-5*x-100))/((361*x**2+190*x+25)*ln(2)**2+(-722*x**3-380*x**2-3850*x-1000)*ln(2)+361*x
**4+190*x**3+3825*x**2+1000*x+10000),x)

[Out]

3*exp((-2*x*exp(16) + 2*exp(16)*log(2))/(-19*x**2 - 5*x + (19*x + 5)*log(2) - 100))

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