3.34.34 \(\int \frac {e^{\frac {\log ^2(x)+(-8 x-4 x^2+2 x \log (3)) \log (x) \log (5+x)+(16 x^2+16 x^3+4 x^4+(-8 x^2-4 x^3) \log (3)+x^2 \log ^2(3)) \log ^2(5+x)}{x^2 \log ^2(5+x)}} (-2 x \log ^2(x)+((10+2 x+8 x^2+4 x^3-2 x^2 \log (3)) \log (x)+(-10-2 x) \log ^2(x)) \log (5+x)+(-40 x-28 x^2-4 x^3+(10 x+2 x^2) \log (3)+(40 x+8 x^2+(-10 x-2 x^2) \log (3)) \log (x)) \log ^2(5+x)+(80 x^3+56 x^4+8 x^5+(-20 x^3-4 x^4) \log (3)) \log ^3(5+x))}{(5 x^3+x^4) \log ^3(5+x)} \, dx\)

Optimal. Leaf size=26 \[ e^{\left (4+2 x-\log (3)-\frac {\log (x)}{x \log (5+x)}\right )^2} \]

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Rubi [F]  time = 141.64, 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 {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}\right ) \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^2 + 16*x^3 + 4*x^4 + (-8*x^2 - 4*x^3
)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2)/(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 - 2*x^2
*Log[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (40*x +
8*x^2 + (-10*x - 2*x^2)*Log[3])*Log[x])*Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log[3])*L
og[5 + x]^3))/((5*x^3 + x^4)*Log[5 + x]^3),x]

[Out]

8*Defer[Int][81^(-2 - x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^(1 + (-8 - 4*
x + Log[9])/(x*Log[5 + x])), x] + 4*(4 - Log[3])*Defer[Int][81^(-2 - x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16)
+ Log[x]^2/(x^2*Log[5 + x]^2))*x^((-8 - 4*x + Log[9])/(x*Log[5 + x])), x] - 2*Defer[Int][(3^(-8 - 4*x)*E^(16*
x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - 2*x*Log[5 + x])/
(x*Log[5 + x]))*Log[x]^2)/((5 + x)*Log[5 + x]^3), x] + 4*Defer[Int][(81^(-2 - x)*E^(16*x + 4*x^2 + 16*(1 + Log
[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-8 - 4*x + Log[9])/(x*Log[5 + x]))*Log[x])/((5 + x)*Log[5 + x]^2)
, x] + 10*Defer[Int][(3^(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-
4*x - 8*(1 - Log[3]/4) - 3*x*Log[5 + x])/(x*Log[5 + x]))*Log[x])/((5 + x)*Log[5 + x]^2), x] + 2*Defer[Int][(3^
(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) -
2*x*Log[5 + x])/(x*Log[5 + x]))*Log[x])/((5 + x)*Log[5 + x]^2), x] + 2*(4 - Log[3])*Defer[Int][(3^(-8 - 4*x)*
E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - x*Log[5 +
x])/(x*Log[5 + x]))*Log[x])/((5 + x)*Log[5 + x]^2), x] - 10*Defer[Int][(3^(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 +
Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - 3*x*Log[5 + x])/(x*Log[5 + x]))*Log
[x]^2)/((5 + x)*Log[5 + x]^2), x] - 2*Defer[Int][(3^(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]
^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - 2*x*Log[5 + x])/(x*Log[5 + x]))*Log[x]^2)/((5 + x)*Log[5
+ x]^2), x] - 2*(4 - Log[3])*Defer[Int][(3^(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*L
og[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - 2*x*Log[5 + x])/(x*Log[5 + x])))/Log[5 + x], x] - 4*Defer[Int][(3^
(-8 - 4*x)*E^(16*x + 4*x^2 + 16*(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) -
x*Log[5 + x])/(x*Log[5 + x])))/Log[5 + x], x] + 2*(4 - Log[3])*Defer[Int][(3^(-8 - 4*x)*E^(16*x + 4*x^2 + 16*
(1 + Log[3]^2/16) + Log[x]^2/(x^2*Log[5 + x]^2))*x^((-4*x - 8*(1 - Log[3]/4) - 2*x*Log[5 + x])/(x*Log[5 + x]))
*Log[x])/Log[5 + x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}\right ) \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{x^3 (5+x) \log ^3(5+x)} \, dx\\ &=\int \frac {2\ 3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} (\log (x)+x (-4-2 x+\log (3)) \log (5+x)) \left (-\left ((5+x) \log (5+x) \left (-1+2 x^2 \log (5+x)\right )\right )-\log (x) (x+(5+x) \log (5+x))\right )}{(5+x) \log ^3(5+x)} \, dx\\ &=2 \int \frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} (\log (x)+x (-4-2 x+\log (3)) \log (5+x)) \left (-\left ((5+x) \log (5+x) \left (-1+2 x^2 \log (5+x)\right )\right )-\log (x) (x+(5+x) \log (5+x))\right )}{(5+x) \log ^3(5+x)} \, dx\\ &=2 \int \left (2\ 3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{3+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} (4+2 x-\log (3))-\frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{1+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \log ^2(x)}{(5+x) \log ^3(5+x)}+\frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \log (x) \left (5+x+2 x^3+4 x^2 \left (1-\frac {\log (3)}{4}\right )-5 \log (x)-x \log (x)\right )}{(5+x) \log ^2(5+x)}+\frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{1+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \left (-2 x-4 \left (1-\frac {\log (3)}{4}\right )+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)\right )}{\log (5+x)}\right ) \, dx\\ &=-\left (2 \int \frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{1+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \log ^2(x)}{(5+x) \log ^3(5+x)} \, dx\right )+2 \int \frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \log (x) \left (5+x+2 x^3+4 x^2 \left (1-\frac {\log (3)}{4}\right )-5 \log (x)-x \log (x)\right )}{(5+x) \log ^2(5+x)} \, dx+2 \int \frac {3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{1+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} \left (-2 x-4 \left (1-\frac {\log (3)}{4}\right )+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)\right )}{\log (5+x)} \, dx+4 \int 3^{-8-4 x} \exp \left (16 x+4 x^2+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{x^2 \log ^2(5+x)}\right ) x^{3+\frac {-4 x-8 \left (1-\frac {\log (3)}{4}\right )-3 x \log (5+x)}{x \log (5+x)}} (4+2 x-\log (3)) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^2 + 16*x^3 + 4*x^4 + (-8*x^2 -
4*x^3)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2)/(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 -
2*x^2*Log[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (
40*x + 8*x^2 + (-10*x - 2*x^2)*Log[3])*Log[x])*Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log
[3])*Log[5 + x]^3))/((5*x^3 + x^4)*Log[5 + x]^3),x]

[Out]

Integrate[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^2 + 16*x^3 + 4*x^4 + (-8*x^2 -
4*x^3)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2)/(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 -
2*x^2*Log[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (
40*x + 8*x^2 + (-10*x - 2*x^2)*Log[3])*Log[x])*Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log
[3])*Log[5 + x]^3))/((5*x^3 + x^4)*Log[5 + x]^3), x]

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fricas [B]  time = 0.56, size = 82, normalized size = 3.15 \begin {gather*} e^{\left (\frac {{\left (4 \, x^{4} + x^{2} \log \relax (3)^{2} + 16 \, x^{3} + 16 \, x^{2} - 4 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (3)\right )} \log \left (x + 5\right )^{2} - 2 \, {\left (2 \, x^{2} - x \log \relax (3) + 4 \, x\right )} \log \left (x + 5\right ) \log \relax (x) + \log \relax (x)^{2}}{x^{2} \log \left (x + 5\right )^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(
2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)*log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x)
)*log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x^3+16*x^2)*log(5+x)^2+(2*x*log(3)-
4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2/log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 3.65, size = 76, normalized size = 2.92 \begin {gather*} e^{\left (4 \, x^{2} - 4 \, x \log \relax (3) + \log \relax (3)^{2} + 16 \, x + \frac {2 \, \log \relax (3) \log \relax (x)}{x \log \left (x + 5\right )} - \frac {4 \, \log \relax (x)}{\log \left (x + 5\right )} - \frac {8 \, \log \relax (x)}{x \log \left (x + 5\right )} + \frac {\log \relax (x)^{2}}{x^{2} \log \left (x + 5\right )^{2}} - 8 \, \log \relax (3) + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(
2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)*log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x)
)*log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x^3+16*x^2)*log(5+x)^2+(2*x*log(3)-
4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2/log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.45, size = 41, normalized size = 1.58

method	result	size
risch	\({\mathrm e}^{\frac {\left (\ln \left (5+x \right ) \ln \relax (3) x -2 x^{2} \ln \left (5+x \right )-4 x \ln \left (5+x \right )+\ln \relax (x )\right )^{2}}{x^{2} \ln \left (5+x \right )^{2}}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^4-20*x^3)*ln(3)+8*x^5+56*x^4+80*x^3)*ln(5+x)^3+(((-2*x^2-10*x)*ln(3)+8*x^2+40*x)*ln(x)+(2*x^2+10*x
)*ln(3)-4*x^3-28*x^2-40*x)*ln(5+x)^2+((-2*x-10)*ln(x)^2+(-2*x^2*ln(3)+4*x^3+8*x^2+2*x+10)*ln(x))*ln(5+x)-2*x*l
n(x)^2)*exp(((x^2*ln(3)^2+(-4*x^3-8*x^2)*ln(3)+4*x^4+16*x^3+16*x^2)*ln(5+x)^2+(2*x*ln(3)-4*x^2-8*x)*ln(x)*ln(5
+x)+ln(x)^2)/x^2/ln(5+x)^2)/(x^4+5*x^3)/ln(5+x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 1.13, size = 74, normalized size = 2.85 \begin {gather*} \frac {1}{6561} \, e^{\left (4 \, x^{2} - 4 \, x \log \relax (3) + \log \relax (3)^{2} + 16 \, x + \frac {2 \, \log \relax (3) \log \relax (x)}{x \log \left (x + 5\right )} - \frac {4 \, \log \relax (x)}{\log \left (x + 5\right )} - \frac {8 \, \log \relax (x)}{x \log \left (x + 5\right )} + \frac {\log \relax (x)^{2}}{x^{2} \log \left (x + 5\right )^{2}} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(
2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)*log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x)
)*log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x^3+16*x^2)*log(5+x)^2+(2*x*log(3)-
4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2/log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="maxima")

[Out]

1/6561*e^(4*x^2 - 4*x*log(3) + log(3)^2 + 16*x + 2*log(3)*log(x)/(x*log(x + 5)) - 4*log(x)/log(x + 5) - 8*log(
x)/(x*log(x + 5)) + log(x)^2/(x^2*log(x + 5)^2) + 16)

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mupad [B]  time = 2.77, size = 83, normalized size = 3.19 \begin {gather*} \frac {x^{\frac {2\,\ln \relax (3)}{x\,\ln \left (x+5\right )}}\,{\mathrm {e}}^{{\ln \relax (3)}^2}\,{\mathrm {e}}^{16\,x}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{\frac {{\ln \relax (x)}^2}{x^2\,{\ln \left (x+5\right )}^2}}\,{\mathrm {e}}^{4\,x^2}}{6561\,3^{4\,x}\,x^{\frac {8}{x\,\ln \left (x+5\right )}}\,x^{\frac {4}{\ln \left (x+5\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((log(x)^2 + log(x + 5)^2*(x^2*log(3)^2 - log(3)*(8*x^2 + 4*x^3) + 16*x^2 + 16*x^3 + 4*x^4) - log(x +
5)*log(x)*(8*x - 2*x*log(3) + 4*x^2))/(x^2*log(x + 5)^2))*(2*x*log(x)^2 - log(x + 5)*(log(x)*(2*x - 2*x^2*log
(3) + 8*x^2 + 4*x^3 + 10) - log(x)^2*(2*x + 10)) - log(x + 5)^3*(80*x^3 - log(3)*(20*x^3 + 4*x^4) + 56*x^4 + 8
*x^5) + log(x + 5)^2*(40*x - log(3)*(10*x + 2*x^2) + 28*x^2 + 4*x^3 - log(x)*(40*x - log(3)*(10*x + 2*x^2) + 8
*x^2))))/(log(x + 5)^3*(5*x^3 + x^4)),x)

[Out]

(x^((2*log(3))/(x*log(x + 5)))*exp(log(3)^2)*exp(16*x)*exp(16)*exp(log(x)^2/(x^2*log(x + 5)^2))*exp(4*x^2))/(6
561*3^(4*x)*x^(8/(x*log(x + 5)))*x^(4/log(x + 5)))

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sympy [B]  time = 3.30, size = 85, normalized size = 3.27 \begin {gather*} e^{\frac {\left (- 4 x^{2} - 8 x + 2 x \log {\relax (3 )}\right ) \log {\relax (x )} \log {\left (x + 5 \right )} + \left (4 x^{4} + 16 x^{3} + x^{2} \log {\relax (3 )}^{2} + 16 x^{2} + \left (- 4 x^{3} - 8 x^{2}\right ) \log {\relax (3 )}\right ) \log {\left (x + 5 \right )}^{2} + \log {\relax (x )}^{2}}{x^{2} \log {\left (x + 5 \right )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**4-20*x**3)*ln(3)+8*x**5+56*x**4+80*x**3)*ln(5+x)**3+(((-2*x**2-10*x)*ln(3)+8*x**2+40*x)*ln(
x)+(2*x**2+10*x)*ln(3)-4*x**3-28*x**2-40*x)*ln(5+x)**2+((-2*x-10)*ln(x)**2+(-2*x**2*ln(3)+4*x**3+8*x**2+2*x+10
)*ln(x))*ln(5+x)-2*x*ln(x)**2)*exp(((x**2*ln(3)**2+(-4*x**3-8*x**2)*ln(3)+4*x**4+16*x**3+16*x**2)*ln(5+x)**2+(
2*x*ln(3)-4*x**2-8*x)*ln(x)*ln(5+x)+ln(x)**2)/x**2/ln(5+x)**2)/(x**4+5*x**3)/ln(5+x)**3,x)

[Out]

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

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