3.57.2 \(\int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} (900+600 x+160 x^2+20 x^3+x^4)+e^{\frac {5 x^2}{30+10 x+x^2}} (-900 x-600 x^2-160 x^3-20 x^4-x^5) \log (x))}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx\)

Optimal. Leaf size=33 \[ -4+e^{-x} \left (e^{-\frac {x^2}{1+\frac {1}{5} (5+x)^2}} x+\log (x)\right ) \]

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Rubi [F]  time = 14.04, 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^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (900 x-300 x^2-740 x^3-190 x^4-19 x^5-x^6+e^{\frac {5 x^2}{30+10 x+x^2}} \left (900+600 x+160 x^2+20 x^3+x^4\right )+e^{\frac {5 x^2}{30+10 x+x^2}} \left (-900 x-600 x^2-160 x^3-20 x^4-x^5\right ) \log (x)\right )}{900 x+600 x^2+160 x^3+20 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x - (5*x^2)/(30 + 10*x + x^2))*(900*x - 300*x^2 - 740*x^3 - 190*x^4 - 19*x^5 - x^6 + E^((5*x^2)/(30 +
10*x + x^2))*(900 + 600*x + 160*x^2 + 20*x^3 + x^4) + E^((5*x^2)/(30 + 10*x + x^2))*(-900*x - 600*x^2 - 160*x
^3 - 20*x^4 - x^5)*Log[x]))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5),x]

[Out]

Log[x]/E^x + Defer[Int][E^(-((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))), x] + 1200*Defer[Int][1/(E^((x*(30 + 15
*x + x^2))/(30 + 10*x + x^2))*(-10 + (2*I)*Sqrt[5] - 2*x)^2), x] - 100*(5 - I*Sqrt[5])*Defer[Int][1/(E^((x*(30
+ 15*x + x^2))/(30 + 10*x + x^2))*(-10 + (2*I)*Sqrt[5] - 2*x)^2), x] - (70*I)*Sqrt[5]*Defer[Int][1/(E^((x*(30
+ 15*x + x^2))/(30 + 10*x + x^2))*(-10 + (2*I)*Sqrt[5] - 2*x)), x] - Defer[Int][x/E^((x*(30 + 15*x + x^2))/(3
0 + 10*x + x^2)), x] - 10*(5 + (9*I)*Sqrt[5])*Defer[Int][1/(E^((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))*(10 -
(2*I)*Sqrt[5] + 2*x)), x] + 1200*Defer[Int][1/(E^((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))*(10 + (2*I)*Sqrt[5]
+ 2*x)^2), x] - 100*(5 + I*Sqrt[5])*Defer[Int][1/(E^((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))*(10 + (2*I)*Sqr
t[5] + 2*x)^2), x] - (70*I)*Sqrt[5]*Defer[Int][1/(E^((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))*(10 + (2*I)*Sqrt
[5] + 2*x)), x] - 10*(5 - (9*I)*Sqrt[5])*Defer[Int][1/(E^((x*(30 + 15*x + x^2))/(30 + 10*x + x^2))*(10 + (2*I)
*Sqrt[5] + 2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-x} \left (\frac {1}{x}-\frac {e^{-\frac {5 x^2}{30+10 x+x^2}} \left (-900+300 x+740 x^2+190 x^3+19 x^4+x^5\right )}{\left (30+10 x+x^2\right )^2}-\log (x)\right ) \, dx\\ &=\int \left (\frac {e^{-x}}{x}-\frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (-900+300 x+740 x^2+190 x^3+19 x^4+x^5\right )}{\left (30+10 x+x^2\right )^2}-e^{-x} \log (x)\right ) \, dx\\ &=\int \frac {e^{-x}}{x} \, dx-\int \frac {e^{-x-\frac {5 x^2}{30+10 x+x^2}} \left (-900+300 x+740 x^2+190 x^3+19 x^4+x^5\right )}{\left (30+10 x+x^2\right )^2} \, dx-\int e^{-x} \log (x) \, dx\\ &=\text {Ei}(-x)+e^{-x} \log (x)-\int \frac {e^{-x}}{x} \, dx-\int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \left (-900+300 x+740 x^2+190 x^3+19 x^4+x^5\right )}{\left (30+10 x+x^2\right )^2} \, dx\\ &=e^{-x} \log (x)-\int \left (-e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}+e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x+\frac {500 e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} (12+x)}{\left (30+10 x+x^2\right )^2}+\frac {50 e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} (-4+x)}{30+10 x+x^2}\right ) \, dx\\ &=e^{-x} \log (x)-50 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} (-4+x)}{30+10 x+x^2} \, dx-500 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} (12+x)}{\left (30+10 x+x^2\right )^2} \, dx+\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \, dx-\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x \, dx\\ &=e^{-x} \log (x)-50 \int \left (\frac {\left (1+\frac {9 i}{\sqrt {5}}\right ) e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10-2 i \sqrt {5}+2 x}+\frac {\left (1-\frac {9 i}{\sqrt {5}}\right ) e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x}\right ) \, dx-500 \int \left (\frac {12 e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (30+10 x+x^2\right )^2}+\frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x}{\left (30+10 x+x^2\right )^2}\right ) \, dx+\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \, dx-\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x \, dx\\ &=e^{-x} \log (x)-500 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x}{\left (30+10 x+x^2\right )^2} \, dx-6000 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (30+10 x+x^2\right )^2} \, dx-\left (10 \left (5-9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x} \, dx-\left (10 \left (5+9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10-2 i \sqrt {5}+2 x} \, dx+\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \, dx-\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x \, dx\\ &=e^{-x} \log (x)-500 \int \left (-\frac {\left (-10+2 i \sqrt {5}\right ) e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10 \left (-10+2 i \sqrt {5}-2 x\right )^2}-\frac {i e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{2 \sqrt {5} \left (-10+2 i \sqrt {5}-2 x\right )}-\frac {\left (-10-2 i \sqrt {5}\right ) e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10 \left (10+2 i \sqrt {5}+2 x\right )^2}-\frac {i e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{2 \sqrt {5} \left (10+2 i \sqrt {5}+2 x\right )}\right ) \, dx-6000 \int \left (-\frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{5 \left (-10+2 i \sqrt {5}-2 x\right )^2}+\frac {i e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10 \sqrt {5} \left (-10+2 i \sqrt {5}-2 x\right )}-\frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{5 \left (10+2 i \sqrt {5}+2 x\right )^2}+\frac {i e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10 \sqrt {5} \left (10+2 i \sqrt {5}+2 x\right )}\right ) \, dx-\left (10 \left (5-9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x} \, dx-\left (10 \left (5+9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10-2 i \sqrt {5}+2 x} \, dx+\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \, dx-\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x \, dx\\ &=e^{-x} \log (x)+1200 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (-10+2 i \sqrt {5}-2 x\right )^2} \, dx+1200 \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (10+2 i \sqrt {5}+2 x\right )^2} \, dx+\left (50 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{-10+2 i \sqrt {5}-2 x} \, dx+\left (50 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x} \, dx-\left (120 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{-10+2 i \sqrt {5}-2 x} \, dx-\left (120 i \sqrt {5}\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x} \, dx-\left (100 \left (5-i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (-10+2 i \sqrt {5}-2 x\right )^2} \, dx-\left (100 \left (5+i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{\left (10+2 i \sqrt {5}+2 x\right )^2} \, dx-\left (10 \left (5-9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10+2 i \sqrt {5}+2 x} \, dx-\left (10 \left (5+9 i \sqrt {5}\right )\right ) \int \frac {e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}}}{10-2 i \sqrt {5}+2 x} \, dx+\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} \, dx-\int e^{-\frac {x \left (30+15 x+x^2\right )}{30+10 x+x^2}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.09, size = 33, normalized size = 1.00 \begin {gather*} e^{-5-x+\frac {50 (3+x)}{30+10 x+x^2}} x+e^{-x} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x - (5*x^2)/(30 + 10*x + x^2))*(900*x - 300*x^2 - 740*x^3 - 190*x^4 - 19*x^5 - x^6 + E^((5*x^2)
/(30 + 10*x + x^2))*(900 + 600*x + 160*x^2 + 20*x^3 + x^4) + E^((5*x^2)/(30 + 10*x + x^2))*(-900*x - 600*x^2 -
160*x^3 - 20*x^4 - x^5)*Log[x]))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5),x]

[Out]

E^(-5 - x + (50*(3 + x))/(30 + 10*x + x^2))*x + Log[x]/E^x

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fricas [B]  time = 0.87, size = 72, normalized size = 2.18 \begin {gather*} x e^{\left (-\frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} + e^{\left (\frac {5 \, x^{2}}{x^{2} + 10 \, x + 30} - \frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*log(x)+(x^4+20*x^3+160*x^2+600*x+900)*
exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^
2/(x^2+10*x+30))/exp(x),x, algorithm="fricas")

[Out]

x*e^(-(x^3 + 15*x^2 + 30*x)/(x^2 + 10*x + 30)) + e^(5*x^2/(x^2 + 10*x + 30) - (x^3 + 15*x^2 + 30*x)/(x^2 + 10*
x + 30))*log(x)

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giac [A]  time = 0.33, size = 35, normalized size = 1.06 \begin {gather*} x e^{\left (-\frac {x^{3} + 15 \, x^{2} + 30 \, x}{x^{2} + 10 \, x + 30}\right )} + e^{\left (-x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*log(x)+(x^4+20*x^3+160*x^2+600*x+900)*
exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^
2/(x^2+10*x+30))/exp(x),x, algorithm="giac")

[Out]

x*e^(-(x^3 + 15*x^2 + 30*x)/(x^2 + 10*x + 30)) + e^(-x)*log(x)

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maple [A]  time = 0.16, size = 33, normalized size = 1.00

method	result	size
risch	\(\ln \relax (x ) {\mathrm e}^{-x}+x \,{\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}\)	\(33\)
default	\({\mathrm e}^{\frac {5 x^{2}}{x^{2}+10 x +30}} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}} \ln \relax (x )+\frac {x^{3} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}+30 x \,{\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}+10 x^{2} {\mathrm e}^{-\frac {x \left (x^{2}+15 x +30\right )}{x^{2}+10 x +30}}}{x^{2}+10 x +30}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*ln(x)+(x^4+20*x^3+160*x^2+600*x+900)*exp(5*x
^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^2/(x^2+
10*x+30))/exp(x),x,method=_RETURNVERBOSE)

[Out]

ln(x)*exp(-x)+x*exp(-x*(x^2+15*x+30)/(x^2+10*x+30))

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maxima [A]  time = 0.54, size = 42, normalized size = 1.27 \begin {gather*} {\left (x e^{\left (\frac {50 \, x}{x^{2} + 10 \, x + 30} + \frac {150}{x^{2} + 10 \, x + 30}\right )} + e^{5} \log \relax (x)\right )} e^{\left (-x - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5-20*x^4-160*x^3-600*x^2-900*x)*exp(5*x^2/(x^2+10*x+30))*log(x)+(x^4+20*x^3+160*x^2+600*x+900)*
exp(5*x^2/(x^2+10*x+30))-x^6-19*x^5-190*x^4-740*x^3-300*x^2+900*x)/(x^5+20*x^4+160*x^3+600*x^2+900*x)/exp(5*x^
2/(x^2+10*x+30))/exp(x),x, algorithm="maxima")

[Out]

(x*e^(50*x/(x^2 + 10*x + 30) + 150/(x^2 + 10*x + 30)) + e^5*log(x))*e^(-x - 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{-\frac {5\,x^2}{x^2+10\,x+30}}\,{\mathrm {e}}^{-x}\,\left (300\,x^2-900\,x+740\,x^3+190\,x^4+19\,x^5+x^6-{\mathrm {e}}^{\frac {5\,x^2}{x^2+10\,x+30}}\,\left (x^4+20\,x^3+160\,x^2+600\,x+900\right )+{\mathrm {e}}^{\frac {5\,x^2}{x^2+10\,x+30}}\,\ln \relax (x)\,\left (x^5+20\,x^4+160\,x^3+600\,x^2+900\,x\right )\right )}{x^5+20\,x^4+160\,x^3+600\,x^2+900\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(5*x^2)/(10*x + x^2 + 30))*exp(-x)*(300*x^2 - 900*x + 740*x^3 + 190*x^4 + 19*x^5 + x^6 - exp((5*x^2
)/(10*x + x^2 + 30))*(600*x + 160*x^2 + 20*x^3 + x^4 + 900) + exp((5*x^2)/(10*x + x^2 + 30))*log(x)*(900*x + 6
00*x^2 + 160*x^3 + 20*x^4 + x^5)))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5),x)

[Out]

int(-(exp(-(5*x^2)/(10*x + x^2 + 30))*exp(-x)*(300*x^2 - 900*x + 740*x^3 + 190*x^4 + 19*x^5 + x^6 - exp((5*x^2
)/(10*x + x^2 + 30))*(600*x + 160*x^2 + 20*x^3 + x^4 + 900) + exp((5*x^2)/(10*x + x^2 + 30))*log(x)*(900*x + 6
00*x^2 + 160*x^3 + 20*x^4 + x^5)))/(900*x + 600*x^2 + 160*x^3 + 20*x^4 + x^5), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**5-20*x**4-160*x**3-600*x**2-900*x)*exp(5*x**2/(x**2+10*x+30))*ln(x)+(x**4+20*x**3+160*x**2+600
*x+900)*exp(5*x**2/(x**2+10*x+30))-x**6-19*x**5-190*x**4-740*x**3-300*x**2+900*x)/(x**5+20*x**4+160*x**3+600*x
**2+900*x)/exp(5*x**2/(x**2+10*x+30))/exp(x),x)

[Out]

Timed out

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