3.12.72 \(\int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} (e^{x/5} (-135 x^4+27 x^5)+540 x^2 \log (25)+e^x (-135 e^{x/5} x^5+180 x^3 \log (25)))}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx\)

Optimal. Leaf size=32 \[ \frac {3 \left (-4+e^{e^x} x\right )}{-e^{x/5}+\frac {4 \log (25)}{3 x^2}} \]

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Rubi [F]  time = 5.65, 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 {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-108*E^(x/5)*x^4 - 1440*x*Log[25] + E^E^x*(E^(x/5)*(-135*x^4 + 27*x^5) + 540*x^2*Log[25] + E^x*(-135*E^(x
/5)*x^5 + 180*x^3*Log[25])))/(45*E^((2*x)/5)*x^4 - 120*E^(x/5)*x^2*Log[25] + 80*Log[25]^2),x]

[Out]

(-256*Log[25]^4*Defer[Int][E^E^x/x^7, x])/27 - (64*Log[25]^3*Defer[Int][E^((5*E^x + x)/5)/x^5, x])/9 - (16*Log
[25]^2*Defer[Int][E^(E^x + (2*x)/5)/x^3, x])/3 - 4*Log[25]*Defer[Int][E^(E^x + (3*x)/5)/x, x] - 3*Defer[Int][E
^(E^x + (4*x)/5)*x, x] - 288*Log[25]*Defer[Int][x/(3*E^(x/5)*x^2 - 4*Log[25])^2, x] - (144*Log[25]*Defer[Int][
x^2/(3*E^(x/5)*x^2 - 4*Log[25])^2, x])/5 + 72*Log[25]*Defer[Int][(E^E^x*x^2)/(3*E^(x/5)*x^2 - 4*Log[25])^2, x]
+ (36*Log[25]*Defer[Int][(E^E^x*x^3)/(3*E^(x/5)*x^2 - 4*Log[25])^2, x])/5 - (1024*Log[25]^5*Defer[Int][E^E^x/
(x^7*(3*E^(x/5)*x^2 - 4*Log[25])), x])/27 - (36*Defer[Int][x^2/(3*E^(x/5)*x^2 - 4*Log[25]), x])/5 - 9*Defer[In
t][(E^E^x*x^2)/(3*E^(x/5)*x^2 - 4*Log[25]), x] + (9*Defer[Int][(E^E^x*x^3)/(3*E^(x/5)*x^2 - 4*Log[25]), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{5 \left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-15 e^{e^x+\frac {4 x}{5}} x-\frac {20 e^{e^x+\frac {3 x}{5}} \log (25)}{x}+\frac {36 x (10+x) \left (-4+e^{e^x} x\right ) \log (25)}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}-\frac {80 e^{e^x+\frac {2 x}{5}} \log ^2(25)}{3 x^3}-\frac {320 e^{\frac {1}{5} \left (5 e^x+x\right )} \log ^3(25)}{9 x^5}-\frac {1280 e^{e^x} \log ^4(25)}{27 x^7}+\frac {-972 x^9-1215 e^{e^x} x^9+243 e^{e^x} x^{10}-5120 e^{e^x} \log ^5(25)}{27 x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )}\right ) \, dx\\ &=\frac {1}{135} \int \frac {-972 x^9-1215 e^{e^x} x^9+243 e^{e^x} x^{10}-5120 e^{e^x} \log ^5(25)}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )} \, dx-3 \int e^{e^x+\frac {4 x}{5}} x \, dx-(4 \log (25)) \int \frac {e^{e^x+\frac {3 x}{5}}}{x} \, dx+\frac {1}{5} (36 \log (25)) \int \frac {x (10+x) \left (-4+e^{e^x} x\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx-\frac {1}{3} \left (16 \log ^2(25)\right ) \int \frac {e^{e^x+\frac {2 x}{5}}}{x^3} \, dx-\frac {1}{9} \left (64 \log ^3(25)\right ) \int \frac {e^{\frac {1}{5} \left (5 e^x+x\right )}}{x^5} \, dx-\frac {1}{27} \left (256 \log ^4(25)\right ) \int \frac {e^{e^x}}{x^7} \, dx\\ &=\frac {1}{135} \int \left (-\frac {972 x^2}{3 e^{x/5} x^2-4 \log (25)}-\frac {1215 e^{e^x} x^2}{3 e^{x/5} x^2-4 \log (25)}+\frac {243 e^{e^x} x^3}{3 e^{x/5} x^2-4 \log (25)}-\frac {5120 e^{e^x} \log ^5(25)}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )}\right ) \, dx-3 \int e^{e^x+\frac {4 x}{5}} x \, dx-(4 \log (25)) \int \frac {e^{e^x+\frac {3 x}{5}}}{x} \, dx+\frac {1}{5} (36 \log (25)) \int \left (\frac {10 x \left (-4+e^{e^x} x\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}+\frac {x^2 \left (-4+e^{e^x} x\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}\right ) \, dx-\frac {1}{3} \left (16 \log ^2(25)\right ) \int \frac {e^{e^x+\frac {2 x}{5}}}{x^3} \, dx-\frac {1}{9} \left (64 \log ^3(25)\right ) \int \frac {e^{\frac {1}{5} \left (5 e^x+x\right )}}{x^5} \, dx-\frac {1}{27} \left (256 \log ^4(25)\right ) \int \frac {e^{e^x}}{x^7} \, dx\\ &=\frac {9}{5} \int \frac {e^{e^x} x^3}{3 e^{x/5} x^2-4 \log (25)} \, dx-3 \int e^{e^x+\frac {4 x}{5}} x \, dx-\frac {36}{5} \int \frac {x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-9 \int \frac {e^{e^x} x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-(4 \log (25)) \int \frac {e^{e^x+\frac {3 x}{5}}}{x} \, dx+\frac {1}{5} (36 \log (25)) \int \frac {x^2 \left (-4+e^{e^x} x\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx+(72 \log (25)) \int \frac {x \left (-4+e^{e^x} x\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx-\frac {1}{3} \left (16 \log ^2(25)\right ) \int \frac {e^{e^x+\frac {2 x}{5}}}{x^3} \, dx-\frac {1}{9} \left (64 \log ^3(25)\right ) \int \frac {e^{\frac {1}{5} \left (5 e^x+x\right )}}{x^5} \, dx-\frac {1}{27} \left (256 \log ^4(25)\right ) \int \frac {e^{e^x}}{x^7} \, dx-\frac {1}{27} \left (1024 \log ^5(25)\right ) \int \frac {e^{e^x}}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )} \, dx\\ &=\frac {9}{5} \int \frac {e^{e^x} x^3}{3 e^{x/5} x^2-4 \log (25)} \, dx-3 \int e^{e^x+\frac {4 x}{5}} x \, dx-\frac {36}{5} \int \frac {x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-9 \int \frac {e^{e^x} x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-(4 \log (25)) \int \frac {e^{e^x+\frac {3 x}{5}}}{x} \, dx+\frac {1}{5} (36 \log (25)) \int \left (-\frac {4 x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}+\frac {e^{e^x} x^3}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}\right ) \, dx+(72 \log (25)) \int \left (-\frac {4 x}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}+\frac {e^{e^x} x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}\right ) \, dx-\frac {1}{3} \left (16 \log ^2(25)\right ) \int \frac {e^{e^x+\frac {2 x}{5}}}{x^3} \, dx-\frac {1}{9} \left (64 \log ^3(25)\right ) \int \frac {e^{\frac {1}{5} \left (5 e^x+x\right )}}{x^5} \, dx-\frac {1}{27} \left (256 \log ^4(25)\right ) \int \frac {e^{e^x}}{x^7} \, dx-\frac {1}{27} \left (1024 \log ^5(25)\right ) \int \frac {e^{e^x}}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )} \, dx\\ &=\frac {9}{5} \int \frac {e^{e^x} x^3}{3 e^{x/5} x^2-4 \log (25)} \, dx-3 \int e^{e^x+\frac {4 x}{5}} x \, dx-\frac {36}{5} \int \frac {x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-9 \int \frac {e^{e^x} x^2}{3 e^{x/5} x^2-4 \log (25)} \, dx-(4 \log (25)) \int \frac {e^{e^x+\frac {3 x}{5}}}{x} \, dx+\frac {1}{5} (36 \log (25)) \int \frac {e^{e^x} x^3}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx-\frac {1}{5} (144 \log (25)) \int \frac {x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx+(72 \log (25)) \int \frac {e^{e^x} x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx-(288 \log (25)) \int \frac {x}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2} \, dx-\frac {1}{3} \left (16 \log ^2(25)\right ) \int \frac {e^{e^x+\frac {2 x}{5}}}{x^3} \, dx-\frac {1}{9} \left (64 \log ^3(25)\right ) \int \frac {e^{\frac {1}{5} \left (5 e^x+x\right )}}{x^5} \, dx-\frac {1}{27} \left (256 \log ^4(25)\right ) \int \frac {e^{e^x}}{x^7} \, dx-\frac {1}{27} \left (1024 \log ^5(25)\right ) \int \frac {e^{e^x}}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 33, normalized size = 1.03 \begin {gather*} -\frac {9 x^2 \left (-4+e^{e^x} x\right )}{3 e^{x/5} x^2-4 \log (25)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-108*E^(x/5)*x^4 - 1440*x*Log[25] + E^E^x*(E^(x/5)*(-135*x^4 + 27*x^5) + 540*x^2*Log[25] + E^x*(-13
5*E^(x/5)*x^5 + 180*x^3*Log[25])))/(45*E^((2*x)/5)*x^4 - 120*E^(x/5)*x^2*Log[25] + 80*Log[25]^2),x]

[Out]

(-9*x^2*(-4 + E^E^x*x))/(3*E^(x/5)*x^2 - 4*Log[25])

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fricas [A]  time = 1.08, size = 31, normalized size = 0.97 \begin {gather*} -\frac {9 \, {\left (x^{3} e^{\left (e^{x}\right )} - 4 \, x^{2}\right )}}{3 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} - 8 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5*x)+1080*x^2*log(5))*exp(exp(x
))-108*x^4*exp(1/5*x)-2880*x*log(5))/(45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm
="fricas")

[Out]

-9*(x^3*e^(e^x) - 4*x^2)/(3*x^2*e^(1/5*x) - 8*log(5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {9 \, {\left (12 \, x^{4} e^{\left (\frac {1}{5} \, x\right )} - {\left (120 \, x^{2} \log \relax (5) + 3 \, {\left (x^{5} - 5 \, x^{4}\right )} e^{\left (\frac {1}{5} \, x\right )} - 5 \, {\left (3 \, x^{5} e^{\left (\frac {1}{5} \, x\right )} - 8 \, x^{3} \log \relax (5)\right )} e^{x}\right )} e^{\left (e^{x}\right )} + 320 \, x \log \relax (5)\right )}}{5 \, {\left (9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} - 48 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \relax (5) + 64 \, \log \relax (5)^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5*x)+1080*x^2*log(5))*exp(exp(x
))-108*x^4*exp(1/5*x)-2880*x*log(5))/(45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm
="giac")

[Out]

integrate(-9/5*(12*x^4*e^(1/5*x) - (120*x^2*log(5) + 3*(x^5 - 5*x^4)*e^(1/5*x) - 5*(3*x^5*e^(1/5*x) - 8*x^3*lo
g(5))*e^x)*e^(e^x) + 320*x*log(5))/(9*x^4*e^(2/5*x) - 48*x^2*e^(1/5*x)*log(5) + 64*log(5)^2), x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-135 x^{5} {\mathrm e}^{\frac {x}{5}}+360 x^{3} \ln \relax (5)\right ) {\mathrm e}^{x}+\left (27 x^{5}-135 x^{4}\right ) {\mathrm e}^{\frac {x}{5}}+1080 x^{2} \ln \relax (5)\right ) {\mathrm e}^{{\mathrm e}^{x}}-108 x^{4} {\mathrm e}^{\frac {x}{5}}-2880 x \ln \relax (5)}{45 x^{4} {\mathrm e}^{\frac {2 x}{5}}-240 x^{2} \ln \relax (5) {\mathrm e}^{\frac {x}{5}}+320 \ln \relax (5)^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-135*x^5*exp(1/5*x)+360*x^3*ln(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5*x)+1080*x^2*ln(5))*exp(exp(x))-108*x
^4*exp(1/5*x)-2880*x*ln(5))/(45*x^4*exp(1/5*x)^2-240*x^2*ln(5)*exp(1/5*x)+320*ln(5)^2),x)

[Out]

int((((-135*x^5*exp(1/5*x)+360*x^3*ln(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5*x)+1080*x^2*ln(5))*exp(exp(x))-108*x
^4*exp(1/5*x)-2880*x*ln(5))/(45*x^4*exp(1/5*x)^2-240*x^2*ln(5)*exp(1/5*x)+320*ln(5)^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {4096}{27} \, \int \frac {e^{\left (e^{x}\right )}}{x^{7}}\,{d x} \log \relax (5)^{4} - \frac {512}{9} \, \int \frac {e^{\left (\frac {1}{5} \, x + e^{x}\right )}}{x^{5}}\,{d x} \log \relax (5)^{3} - \frac {64}{3} \, \int \frac {e^{\left (\frac {2}{5} \, x + e^{x}\right )}}{x^{3}}\,{d x} \log \relax (5)^{2} - 8 \, \int \frac {e^{\left (\frac {3}{5} \, x + e^{x}\right )}}{x}\,{d x} \log \relax (5) - \frac {9 \, {\left (x^{3} e^{\left (e^{x}\right )} - 4 \, x^{2}\right )}}{3 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} - 8 \, \log \relax (5)} - 3 \, \int x e^{\left (\frac {4}{5} \, x + e^{x}\right )}\,{d x} + \frac {9}{5} \, \int \frac {5 \, {\left (243 \, x^{10} e^{x} - 32768 \, \log \relax (5)^{5}\right )} e^{\left (e^{x}\right )}}{243 \, {\left (3 \, x^{9} e^{\left (\frac {1}{5} \, x\right )} - 8 \, x^{7} \log \relax (5)\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5*x)+1080*x^2*log(5))*exp(exp(x
))-108*x^4*exp(1/5*x)-2880*x*log(5))/(45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm
="maxima")

[Out]

-4096/27*integrate(e^(e^x)/x^7, x)*log(5)^4 - 512/9*integrate(e^(1/5*x + e^x)/x^5, x)*log(5)^3 - 64/3*integrat
e(e^(2/5*x + e^x)/x^3, x)*log(5)^2 - 8*integrate(e^(3/5*x + e^x)/x, x)*log(5) - 9*(x^3*e^(e^x) - 4*x^2)/(3*x^2
*e^(1/5*x) - 8*log(5)) - 3*integrate(x*e^(4/5*x + e^x), x) + 9/5*integrate(5/243*(243*x^10*e^x - 32768*log(5)^
5)*e^(e^x)/(3*x^9*e^(1/5*x) - 8*x^7*log(5)), x)

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mupad [B]  time = 1.16, size = 28, normalized size = 0.88 \begin {gather*} \frac {9\,x^2\,\left (x\,{\mathrm {e}}^{{\mathrm {e}}^x}-4\right )}{8\,\ln \relax (5)-3\,x^2\,{\mathrm {e}}^{x/5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2880*x*log(5) + 108*x^4*exp(x/5) + exp(exp(x))*(exp(x/5)*(135*x^4 - 27*x^5) - 1080*x^2*log(5) + exp(x)*(
135*x^5*exp(x/5) - 360*x^3*log(5))))/(45*x^4*exp((2*x)/5) + 320*log(5)^2 - 240*x^2*exp(x/5)*log(5)),x)

[Out]

(9*x^2*(x*exp(exp(x)) - 4))/(8*log(5) - 3*x^2*exp(x/5))

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sympy [A]  time = 0.27, size = 44, normalized size = 1.38 \begin {gather*} - \frac {9 x^{3} e^{e^{x}}}{3 x^{2} e^{\frac {x}{5}} - 8 \log {\relax (5 )}} + \frac {36 x^{2}}{3 x^{2} e^{\frac {x}{5}} - 8 \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-135*x**5*exp(1/5*x)+360*x**3*ln(5))*exp(x)+(27*x**5-135*x**4)*exp(1/5*x)+1080*x**2*ln(5))*exp(ex
p(x))-108*x**4*exp(1/5*x)-2880*x*ln(5))/(45*x**4*exp(1/5*x)**2-240*x**2*ln(5)*exp(1/5*x)+320*ln(5)**2),x)

[Out]

-9*x**3*exp(exp(x))/(3*x**2*exp(x/5) - 8*log(5)) + 36*x**2/(3*x**2*exp(x/5) - 8*log(5))

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