3.64.23 \(\int \frac {e^{\frac {-2 x+2 x^4+e^x (-x+x^4)+(-x+x^4) \log (\log (5))}{35+7 x}} (175+60 x+7 x^2+40 x^4+6 x^5+e^x (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6)+(-5 x+20 x^4+3 x^5) \log (\log (5)))}{175+70 x+7 x^2} \, dx\)

Optimal. Leaf size=28 \[ e^{\frac {\left (-x+x^4\right ) \left (2+e^x+\log (\log (5))\right )}{7 (5+x)}} x \]

________________________________________________________________________________________

Rubi [F]  time = 27.42, 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 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{175+70 x+7 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-2*x + 2*x^4 + E^x*(-x + x^4) + (-x + x^4)*Log[Log[5]])/(35 + 7*x))*(175 + 60*x + 7*x^2 + 40*x^4 + 6*
x^5 + E^x*(-5*x - 5*x^2 - x^3 + 20*x^4 + 8*x^5 + x^6) + (-5*x + 20*x^4 + 3*x^5)*Log[Log[5]]))/(175 + 70*x + 7*
x^2),x]

[Out]

90*Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x)), x] + Defer[Int][E^((x*(-1 + x^3
)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x)), x] - (101*Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[
5]]/2)))/(35 + 7*x))*x, x])/7 + (50*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x,
x])/7 + (25*Log[Log[5]]*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x, x])/7 + (15*
Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x^2, x])/7 - (20*Defer[Int][E^((x*(
-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x^2, x])/7 - (10*Log[Log[5]]*Defer[Int][E^((x*(-1 + x^3)*
(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x^2, x])/7 - (2*Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[L
og[5]]/2)))/(35 + 7*x))*x^3, x])/7 + (6*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))
*x^3, x])/7 + (3*Log[Log[5]]*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x^3, x])/7
+ Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))*x^4, x]/7 + 450*Defer[Int][E^(x
+ (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))/(5 + x)^2, x] + 900*Defer[Int][E^((x*(-1 + x^3)*(E^
x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))/(5 + x)^2, x] + 450*Log[Log[5]]*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(
1 + Log[Log[5]]/2)))/(35 + 7*x))/(5 + x)^2, x] - 540*Defer[Int][E^(x + (x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]
/2)))/(35 + 7*x))/(5 + x), x] - 180*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))/(5
+ x), x] - 90*Log[Log[5]]*Defer[Int][E^((x*(-1 + x^3)*(E^x + 2*(1 + Log[Log[5]]/2)))/(35 + 7*x))/(5 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{7 (5+x)^2} \, dx\\ &=\frac {1}{7} \int \frac {\exp \left (\frac {-2 x+2 x^4+e^x \left (-x+x^4\right )+\left (-x+x^4\right ) \log (\log (5))}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{(5+x)^2} \, dx\\ &=\frac {1}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) \left (175+60 x+7 x^2+40 x^4+6 x^5+e^x \left (-5 x-5 x^2-x^3+20 x^4+8 x^5+x^6\right )+\left (-5 x+20 x^4+3 x^5\right ) \log (\log (5))\right )}{(5+x)^2} \, dx\\ &=\frac {1}{7} \int \left (\frac {175 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right )}{(5+x)^2}+\frac {60 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x}{(5+x)^2}+\frac {7 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^2}{(5+x)^2}+\frac {40 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^4}{(5+x)^2}+\frac {6 \exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^5}{(5+x)^2}+\frac {\exp \left (x+\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5-5 x-x^2+20 x^3+8 x^4+x^5\right )}{(5+x)^2}+\frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5+20 x^3+3 x^4\right ) \log (\log (5))}{(5+x)^2}\right ) \, dx\\ &=\frac {1}{7} \int \frac {\exp \left (x+\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5-5 x-x^2+20 x^3+8 x^4+x^5\right )}{(5+x)^2} \, dx+\frac {6}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^5}{(5+x)^2} \, dx+\frac {40}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^4}{(5+x)^2} \, dx+\frac {60}{7} \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x}{(5+x)^2} \, dx+25 \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right )}{(5+x)^2} \, dx+\frac {1}{7} \log (\log (5)) \int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x \left (-5+20 x^3+3 x^4\right )}{(5+x)^2} \, dx+\int \frac {\exp \left (\frac {x \left (-1+x^3\right ) \left (e^x+2 \left (1+\frac {1}{2} \log (\log (5))\right )\right )}{35+7 x}\right ) x^2}{(5+x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 42, normalized size = 1.50 \begin {gather*} e^{\frac {\left (2+e^x\right ) x \left (-1+x^3\right )}{7 (5+x)}} x \log ^{\frac {x \left (-1+x^3\right )}{7 (5+x)}}(5) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-2*x + 2*x^4 + E^x*(-x + x^4) + (-x + x^4)*Log[Log[5]])/(35 + 7*x))*(175 + 60*x + 7*x^2 + 40*x^
4 + 6*x^5 + E^x*(-5*x - 5*x^2 - x^3 + 20*x^4 + 8*x^5 + x^6) + (-5*x + 20*x^4 + 3*x^5)*Log[Log[5]]))/(175 + 70*
x + 7*x^2),x]

[Out]

E^(((2 + E^x)*x*(-1 + x^3))/(7*(5 + x)))*x*Log[5]^((x*(-1 + x^3))/(7*(5 + x)))

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 40, normalized size = 1.43 \begin {gather*} x e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \relax (5)\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)
*exp(((x^4-x)*log(log(5))+(x^4-x)*exp(x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="fricas")

[Out]

x*e^(1/7*(2*x^4 + (x^4 - x)*e^x + (x^4 - x)*log(log(5)) - 2*x)/(x + 5))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)
*exp(((x^4-x)*log(log(5))+(x^4-x)*exp(x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="giac")

[Out]

undef

________________________________________________________________________________________

maple [A]  time = 0.46, size = 28, normalized size = 1.00

method	result	size
risch	\(x \,{\mathrm e}^{\frac {\left (2+{\mathrm e}^{x}+\ln \left (\ln \relax (5)\right )\right ) \left (x^{2}+x +1\right ) x \left (x -1\right )}{7 x +35}}\)	\(28\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \relax (5)\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}+5 x \,{\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (\ln \relax (5)\right )+\left (x^{4}-x \right ) {\mathrm e}^{x}+2 x^{4}-2 x}{7 x +35}}}{5+x}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^5+20*x^4-5*x)*ln(ln(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)*exp(((x
^4-x)*ln(ln(5))+(x^4-x)*exp(x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x,method=_RETURNVERBOSE)

[Out]

x*exp(1/7*x*(x-1)*(x^2+x+1)*(2+exp(x)+ln(ln(5)))/(5+x))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{7} \, \int \frac {{\left (6 \, x^{5} + 40 \, x^{4} + 7 \, x^{2} + {\left (x^{6} + 8 \, x^{5} + 20 \, x^{4} - x^{3} - 5 \, x^{2} - 5 \, x\right )} e^{x} + {\left (3 \, x^{5} + 20 \, x^{4} - 5 \, x\right )} \log \left (\log \relax (5)\right ) + 60 \, x + 175\right )} e^{\left (\frac {2 \, x^{4} + {\left (x^{4} - x\right )} e^{x} + {\left (x^{4} - x\right )} \log \left (\log \relax (5)\right ) - 2 \, x}{7 \, {\left (x + 5\right )}}\right )}}{x^{2} + 10 \, x + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5+20*x^4-5*x)*log(log(5))+(x^6+8*x^5+20*x^4-x^3-5*x^2-5*x)*exp(x)+6*x^5+40*x^4+7*x^2+60*x+175)
*exp(((x^4-x)*log(log(5))+(x^4-x)*exp(x)+2*x^4-2*x)/(7*x+35))/(7*x^2+70*x+175),x, algorithm="maxima")

[Out]

1/7*integrate((6*x^5 + 40*x^4 + 7*x^2 + (x^6 + 8*x^5 + 20*x^4 - x^3 - 5*x^2 - 5*x)*e^x + (3*x^5 + 20*x^4 - 5*x
)*log(log(5)) + 60*x + 175)*e^(1/7*(2*x^4 + (x^4 - x)*e^x + (x^4 - x)*log(log(5)) - 2*x)/(x + 5))/(x^2 + 10*x
+ 25), x)

________________________________________________________________________________________

mupad [B]  time = 4.51, size = 73, normalized size = 2.61 \begin {gather*} \frac {x\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{\frac {2\,x^4}{7\,x+35}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{7\,x+35}}\,{\mathrm {e}}^{-\frac {2\,x}{7\,x+35}}}{{\ln \relax (5)}^{\frac {x-x^4}{7\,\left (x+5\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*x + log(log(5))*(x - x^4) + exp(x)*(x - x^4) - 2*x^4)/(7*x + 35))*(60*x - exp(x)*(5*x + 5*x^2 + x
^3 - 20*x^4 - 8*x^5 - x^6) + 7*x^2 + 40*x^4 + 6*x^5 + log(log(5))*(20*x^4 - 5*x + 3*x^5) + 175))/(70*x + 7*x^2
+ 175),x)

[Out]

(x*exp((x^4*exp(x))/(7*x + 35))*exp((2*x^4)/(7*x + 35))*exp(-(x*exp(x))/(7*x + 35))*exp(-(2*x)/(7*x + 35)))/lo
g(5)^((x - x^4)/(7*(x + 5)))

________________________________________________________________________________________

sympy [A]  time = 17.06, size = 34, normalized size = 1.21 \begin {gather*} x e^{\frac {2 x^{4} - 2 x + \left (x^{4} - x\right ) e^{x} + \left (x^{4} - x\right ) \log {\left (\log {\relax (5 )} \right )}}{7 x + 35}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**5+20*x**4-5*x)*ln(ln(5))+(x**6+8*x**5+20*x**4-x**3-5*x**2-5*x)*exp(x)+6*x**5+40*x**4+7*x**2+6
0*x+175)*exp(((x**4-x)*ln(ln(5))+(x**4-x)*exp(x)+2*x**4-2*x)/(7*x+35))/(7*x**2+70*x+175),x)

[Out]

x*exp((2*x**4 - 2*x + (x**4 - x)*exp(x) + (x**4 - x)*log(log(5)))/(7*x + 35))

________________________________________________________________________________________