3.16.59 \(\int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} (2500+500 x+20 x^3-4 x^4)+e^6 (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5))}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 (4 x^3+2 x^4))} \, dx\)

Optimal. Leaf size=32 \[ \log \left (\frac {5}{\left (-e^{\left (x-\frac {5 (5+x)}{x}\right )^2}+\frac {x}{2+e^6+x}\right )^2}\right ) \]

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Rubi [F]  time = 5.02, 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 {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^3 + 2*E^6*x^3 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^2)*(10000 + 12000*x + 4500*x^2 + 580*x^3 +
64*x^4 + 4*x^5 - 4*x^6 + E^12*(2500 + 500*x + 20*x^3 - 4*x^4) + E^6*(10000 + 7000*x + 1000*x^2 + 80*x^3 + 24*
x^4 - 8*x^5)))/(-2*x^4 - E^6*x^4 - x^5 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^2)*(4*x^3 + E^12*x^3 + 4*x
^4 + x^5 + E^6*(4*x^3 + 2*x^4))),x]

[Out]

(-2*(25 + 5*x - x^2)^2)/x^2 + 4*Defer[Int][x^2/(-2*E^((-25 - 5*x + x^2)^2/x^2)*(1 + E^6/2) + x - E^((-25 - 5*x
+ x^2)^2/x^2)*x), x] + 2500*Defer[Int][1/(x^2*(2*E^((-25 - 5*x + x^2)^2/x^2)*(1 + E^6/2) - x + E^((-25 - 5*x
+ x^2)^2/x^2)*x)), x] + 500*Defer[Int][1/(x*(2*E^((-25 - 5*x + x^2)^2/x^2)*(1 + E^6/2) - x + E^((-25 - 5*x + x
^2)^2/x^2)*x)), x] + 20*Defer[Int][x/(2*E^((-25 - 5*x + x^2)^2/x^2)*(1 + E^6/2) - x + E^((-25 - 5*x + x^2)^2/x
^2)*x), x] + 2*(2 + E^6)*Defer[Int][1/((2 + E^6 + x)*(2*E^((-25 - 5*x + x^2)^2/x^2)*(1 + E^6/2) - x + E^((-25
- 5*x + x^2)^2/x^2)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{\left (-2-e^6\right ) x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx\\ &=\int \frac {\left (4+2 e^6\right ) x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{\left (-2-e^6\right ) x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx\\ &=\int \frac {-2 \left (2+e^6\right ) x^3+4 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (2+e^6+x\right )^2 \left (-625-125 x-5 x^3+x^4\right )}{x^3 \left (\left (2+e^6\right ) x+x^2-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (2+e^6+x\right )^2\right )} \, dx\\ &=\int \left (-\frac {4 \left (25+x^2\right ) \left (-25-5 x+x^2\right )}{x^3}+\frac {2 \left (2500 \left (1+\frac {e^6}{2}\right )+1750 \left (1+\frac {e^6}{7}\right ) x+252 \left (1+\frac {e^6}{252}\right ) x^2+20 \left (1+\frac {e^6}{2}\right ) x^3+6 \left (1-\frac {e^6}{3}\right ) x^4-2 x^5\right )}{x^2 \left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}\right ) \, dx\\ &=2 \int \frac {2500 \left (1+\frac {e^6}{2}\right )+1750 \left (1+\frac {e^6}{7}\right ) x+252 \left (1+\frac {e^6}{252}\right ) x^2+20 \left (1+\frac {e^6}{2}\right ) x^3+6 \left (1-\frac {e^6}{3}\right ) x^4-2 x^5}{x^2 \left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx-4 \int \frac {\left (25+x^2\right ) \left (-25-5 x+x^2\right )}{x^3} \, dx\\ &=-\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+2 \int \frac {2500+1750 x+252 x^2+20 x^3+6 x^4-2 x^5+e^6 \left (1250+250 x+x^2+10 x^3-2 x^4\right )}{x^2 \left (2+e^6+x\right ) \left (e^{6+\frac {\left (-25-5 x+x^2\right )^2}{x^2}}-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} (2+x)\right )} \, dx\\ &=-\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+2 \int \left (\frac {2 x^2}{-2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )+x-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x}+\frac {1250}{x^2 \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}+\frac {250}{x \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}+\frac {10 x}{2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x}+\frac {2+e^6}{\left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )}\right ) \, dx\\ &=-\frac {2 \left (25+5 x-x^2\right )^2}{x^2}+4 \int \frac {x^2}{-2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )+x-e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x} \, dx+20 \int \frac {x}{2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x} \, dx+500 \int \frac {1}{x \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx+2500 \int \frac {1}{x^2 \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx+\left (2 \left (2+e^6\right )\right ) \int \frac {1}{\left (2+e^6+x\right ) \left (2 e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )-x+e^{\frac {\left (-25-5 x+x^2\right )^2}{x^2}} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.21, size = 82, normalized size = 2.56 \begin {gather*} 2 \log \left (2+e^6+x\right )-2 \log \left (2 e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}+e^{6+\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}-e^{25} x+e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^3 + 2*E^6*x^3 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^2)*(10000 + 12000*x + 4500*x^2 + 580
*x^3 + 64*x^4 + 4*x^5 - 4*x^6 + E^12*(2500 + 500*x + 20*x^3 - 4*x^4) + E^6*(10000 + 7000*x + 1000*x^2 + 80*x^3
+ 24*x^4 - 8*x^5)))/(-2*x^4 - E^6*x^4 - x^5 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^2)*(4*x^3 + E^12*x^3
+ 4*x^4 + x^5 + E^6*(4*x^3 + 2*x^4))),x]

[Out]

2*Log[2 + E^6 + x] - 2*Log[2*E^(625/x^2 + 250/x - 10*x + x^2) + E^(6 + 625/x^2 + 250/x - 10*x + x^2) - E^25*x
+ E^(625/x^2 + 250/x - 10*x + x^2)*x]

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fricas [A]  time = 0.96, size = 44, normalized size = 1.38 \begin {gather*} -2 \, \log \left (\frac {{\left (x + e^{6} + 2\right )} e^{\left (\frac {x^{4} - 10 \, x^{3} - 25 \, x^{2} + 250 \, x + 625}{x^{2}}\right )} - x}{x + e^{6} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*
x^5+64*x^4+580*x^3+4500*x^2+12000*x+10000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*
exp(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4
),x, algorithm="fricas")

[Out]

-2*log(((x + e^6 + 2)*e^((x^4 - 10*x^3 - 25*x^2 + 250*x + 625)/x^2) - x)/(x + e^6 + 2))

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giac [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((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*
x^5+64*x^4+580*x^3+4500*x^2+12000*x+10000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*
exp(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4
),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.45, size = 71, normalized size = 2.22

method	result	size
risch	\(-2 x^{2}+20 x +\frac {-500 x -1250}{x^{2}}+\frac {2 x^{4}-20 x^{3}-50 x^{2}+500 x +1250}{x^{2}}-2 \ln \left ({\mathrm e}^{\frac {\left (x^{2}-5 x -25\right )^{2}}{x^{2}}}-\frac {x}{{\mathrm e}^{6}+2+x}\right )\)	\(71\)
norman	\(2 \ln \left ({\mathrm e}^{6}+2+x \right )-2 \ln \left ({\mathrm e}^{6} {\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}+{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} x -x +2 \,{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}\right )\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*x^5+64
*x^4+580*x^3+4500*x^2+12000*x+10000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*exp(3)
^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4),x,me
thod=_RETURNVERBOSE)

[Out]

-2*x^2+20*x+(-500*x-1250)/x^2+2*(x^4-10*x^3-25*x^2+250*x+625)/x^2-2*ln(exp((x^2-5*x-25)^2/x^2)-x/(exp(6)+2+x))

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maxima [B]  time = 0.56, size = 70, normalized size = 2.19 \begin {gather*} -\frac {2 \, {\left (x^{3} - 10 \, x^{2} + 250\right )}}{x} - 2 \, \log \left (\frac {{\left ({\left (x + e^{6} + 2\right )} e^{\left (x^{2} + \frac {250}{x} + \frac {625}{x^{2}}\right )} - x e^{\left (10 \, x + 25\right )}\right )} e^{\left (-x^{2} - \frac {250}{x}\right )}}{x + e^{6} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*
x^5+64*x^4+580*x^3+4500*x^2+12000*x+10000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*
exp(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4
),x, algorithm="maxima")

[Out]

-2*(x^3 - 10*x^2 + 250)/x - 2*log(((x + e^6 + 2)*e^(x^2 + 250/x + 625/x^2) - x*e^(10*x + 25))*e^(-x^2 - 250/x)
/(x + e^6 + 2))

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mupad [B]  time = 1.75, size = 95, normalized size = 2.97 \begin {gather*} -2\,\ln \left (\frac {2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}-x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{x^2}\,{\mathrm {e}}^6\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}+x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}}{2\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{31}+x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((250*x - 25*x^2 - 10*x^3 + x^4 + 625)/x^2)*(12000*x + exp(12)*(500*x + 20*x^3 - 4*x^4 + 2500) + exp(
6)*(7000*x + 1000*x^2 + 80*x^3 + 24*x^4 - 8*x^5 + 10000) + 4500*x^2 + 580*x^3 + 64*x^4 + 4*x^5 - 4*x^6 + 10000
) + 2*x^3*exp(6) + 4*x^3)/(x^4*exp(6) - exp((250*x - 25*x^2 - 10*x^3 + x^4 + 625)/x^2)*(exp(6)*(4*x^3 + 2*x^4)
+ x^3*exp(12) + 4*x^3 + 4*x^4 + x^5) + 2*x^4 + x^5),x)

[Out]

-2*log((2*exp(x^2)*exp(250/x)*exp(625/x^2) - x*exp(10*x)*exp(25) + exp(x^2)*exp(6)*exp(250/x)*exp(625/x^2) + x
*exp(x^2)*exp(250/x)*exp(625/x^2))/(2*exp(10*x)*exp(25) + exp(10*x)*exp(31) + x*exp(10*x)*exp(25)))

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sympy [A]  time = 1.20, size = 36, normalized size = 1.12 \begin {gather*} - 2 \log {\left (- \frac {x}{x + 2 + e^{6}} + e^{\frac {x^{4} - 10 x^{3} - 25 x^{2} + 250 x + 625}{x^{2}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**4+20*x**3+500*x+2500)*exp(3)**4+(-8*x**5+24*x**4+80*x**3+1000*x**2+7000*x+10000)*exp(3)**2-
4*x**6+4*x**5+64*x**4+580*x**3+4500*x**2+12000*x+10000)*exp((x**4-10*x**3-25*x**2+250*x+625)/x**2)+2*x**3*exp(
3)**2+4*x**3)/((x**3*exp(3)**4+(2*x**4+4*x**3)*exp(3)**2+x**5+4*x**4+4*x**3)*exp((x**4-10*x**3-25*x**2+250*x+6
25)/x**2)-x**4*exp(3)**2-x**5-2*x**4),x)

[Out]

-2*log(-x/(x + 2 + exp(6)) + exp((x**4 - 10*x**3 - 25*x**2 + 250*x + 625)/x**2))

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