3.47.62 \(\int \frac {256\ 25^{108+36 x+4 x^2} (e^{-x} x)^{108+36 x+4 x^2} (108-72 x-32 x^2-4 x^3+(36 x+8 x^2) \log (25 e^{-x} x))}{x} \, dx\)

Optimal. Leaf size=29 \[ 3+e^{4 \left (\log (4)+\left (2-x+(5+x)^2\right ) \log \left (25 e^{-x} x\right )\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 4.03, 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 {256\ 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \left (108-72 x-32 x^2-4 x^3+\left (36 x+8 x^2\right ) \log \left (25 e^{-x} x\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(256*25^(108 + 36*x + 4*x^2)*(x/E^x)^(108 + 36*x + 4*x^2)*(108 - 72*x - 32*x^2 - 4*x^3 + (36*x + 8*x^2)*Lo
g[(25*x)/E^x]))/x,x]

[Out]

-18432*Defer[Int][25^(108 + 36*x + 4*x^2)*(x/E^x)^(108 + 36*x + 4*x^2), x] + 9216*Log[(25*x)/E^x]*Defer[Int][2
5^(108 + 36*x + 4*x^2)*(x/E^x)^(108 + 36*x + 4*x^2), x] + 27648*Defer[Int][(25^(108 + 36*x + 4*x^2)*(x/E^x)^(1
08 + 36*x + 4*x^2))/x, x] - 8192*Defer[Int][25^(108 + 36*x + 4*x^2)*x*(x/E^x)^(108 + 36*x + 4*x^2), x] + 2048*
Log[(25*x)/E^x]*Defer[Int][25^(108 + 36*x + 4*x^2)*x*(x/E^x)^(108 + 36*x + 4*x^2), x] - 1024*Defer[Int][25^(10
8 + 36*x + 4*x^2)*x^2*(x/E^x)^(108 + 36*x + 4*x^2), x] + 9216*Defer[Int][Defer[Int][25^(4*(27 + 9*x + x^2))*(x
/E^x)^(4*(27 + 9*x + x^2)), x], x] - 9216*Defer[Int][Defer[Int][25^(4*(27 + 9*x + x^2))*(x/E^x)^(4*(27 + 9*x +
x^2)), x]/x, x] + 2048*Defer[Int][Defer[Int][25^(4*(27 + 9*x + x^2))*x*(x/E^x)^(4*(27 + 9*x + x^2)), x], x] -
2048*Defer[Int][Defer[Int][25^(4*(27 + 9*x + x^2))*x*(x/E^x)^(4*(27 + 9*x + x^2)), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=256 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \left (108-72 x-32 x^2-4 x^3+\left (36 x+8 x^2\right ) \log \left (25 e^{-x} x\right )\right )}{x} \, dx\\ &=256 \int \left (-\frac {4\ 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \left (-27+18 x+8 x^2+x^3\right )}{x}+4\ 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} (9+2 x) \log \left (25 e^{-x} x\right )\right ) \, dx\\ &=-\left (1024 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \left (-27+18 x+8 x^2+x^3\right )}{x} \, dx\right )+1024 \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} (9+2 x) \log \left (25 e^{-x} x\right ) \, dx\\ &=-\left (1024 \int \left (18\ 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}-\frac {27\ 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}}{x}+8\ 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2}+25^{108+36 x+4 x^2} x^2 \left (e^{-x} x\right )^{108+36 x+4 x^2}\right ) \, dx\right )-1024 \int \frac {(1-x) \left (9 \int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx+2 \int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx\right )}{x} \, dx+\left (2048 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+\left (9216 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\\ &=-\left (1024 \int 25^{108+36 x+4 x^2} x^2 \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\right )-1024 \int \left (-\frac {9 (-1+x) \int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x}-\frac {2 (-1+x) \int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x}\right ) \, dx-8192 \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx-18432 \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+27648 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}}{x} \, dx+\left (2048 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+\left (9216 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\\ &=-\left (1024 \int 25^{108+36 x+4 x^2} x^2 \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\right )+2048 \int \frac {(-1+x) \int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x} \, dx-8192 \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+9216 \int \frac {(-1+x) \int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x} \, dx-18432 \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+27648 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}}{x} \, dx+\left (2048 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+\left (9216 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\\ &=-\left (1024 \int 25^{108+36 x+4 x^2} x^2 \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\right )+2048 \int \left (\int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx-\frac {\int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x}\right ) \, dx-8192 \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+9216 \int \left (\int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx-\frac {\int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x}\right ) \, dx-18432 \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+27648 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}}{x} \, dx+\left (2048 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+\left (9216 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\\ &=-\left (1024 \int 25^{108+36 x+4 x^2} x^2 \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\right )+2048 \int \left (\int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx\right ) \, dx-2048 \int \frac {\int 25^{4 \left (27+9 x+x^2\right )} x \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x} \, dx-8192 \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+9216 \int \left (\int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx\right ) \, dx-9216 \int \frac {\int 25^{4 \left (27+9 x+x^2\right )} \left (e^{-x} x\right )^{4 \left (27+9 x+x^2\right )} \, dx}{x} \, dx-18432 \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+27648 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2}}{x} \, dx+\left (2048 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} x \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx+\left (9216 \log \left (25 e^{-x} x\right )\right ) \int 25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [F]  time = 5.27, size = 72, normalized size = 2.48 \begin {gather*} 256 \int \frac {25^{108+36 x+4 x^2} \left (e^{-x} x\right )^{108+36 x+4 x^2} \left (108-72 x-32 x^2-4 x^3+\left (36 x+8 x^2\right ) \log \left (25 e^{-x} x\right )\right )}{x} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256*25^(108 + 36*x + 4*x^2)*(x/E^x)^(108 + 36*x + 4*x^2)*(108 - 72*x - 32*x^2 - 4*x^3 + (36*x + 8*x
^2)*Log[(25*x)/E^x]))/x,x]

[Out]

256*Integrate[(25^(108 + 36*x + 4*x^2)*(x/E^x)^(108 + 36*x + 4*x^2)*(108 - 72*x - 32*x^2 - 4*x^3 + (36*x + 8*x
^2)*Log[(25*x)/E^x]))/x, x]

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 24, normalized size = 0.83 \begin {gather*} e^{\left (4 \, {\left (x^{2} + 9 \, x + 27\right )} \log \left (25 \, x e^{\left (-x\right )}\right ) + 8 \, \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+36*x)*log(25*x/exp(x))-4*x^3-32*x^2-72*x+108)*exp((4*x^2+36*x+108)*log(25*x/exp(x))+8*log(2)
)/x,x, algorithm="fricas")

[Out]

e^(4*(x^2 + 9*x + 27)*log(25*x*e^(-x)) + 8*log(2))

________________________________________________________________________________________

giac [A]  time = 0.31, size = 40, normalized size = 1.38 \begin {gather*} e^{\left (4 \, x^{2} \log \left (25 \, x e^{\left (-x\right )}\right ) + 36 \, x \log \left (25 \, x e^{\left (-x\right )}\right ) + 8 \, \log \relax (2) + 108 \, \log \left (25 \, x e^{\left (-x\right )}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+36*x)*log(25*x/exp(x))-4*x^3-32*x^2-72*x+108)*exp((4*x^2+36*x+108)*log(25*x/exp(x))+8*log(2)
)/x,x, algorithm="giac")

[Out]

e^(4*x^2*log(25*x*e^(-x)) + 36*x*log(25*x*e^(-x)) + 8*log(2) + 108*log(25*x*e^(-x)))

________________________________________________________________________________________

maple [C]  time = 0.11, size = 112, normalized size = 3.86

method	result	size
risch	\(256 \,{\mathrm e}^{2 \left (x^{2}+9 x +27\right ) \left (-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+2 \ln \relax (x )-2 \ln \left ({\mathrm e}^{x}\right )+4 \ln \relax (5)\right )}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2+36*x)*ln(25*x/exp(x))-4*x^3-32*x^2-72*x+108)*exp((4*x^2+36*x+108)*ln(25*x/exp(x))+8*ln(2))/x,x,met
hod=_RETURNVERBOSE)

[Out]

256*exp(2*(x^2+9*x+27)*(-I*Pi*csgn(I*x*exp(-x))^3+I*Pi*csgn(I*x*exp(-x))^2*csgn(I*x)+I*Pi*csgn(I*x*exp(-x))^2*
csgn(I*exp(-x))-I*Pi*csgn(I*x*exp(-x))*csgn(I*x)*csgn(I*exp(-x))+2*ln(x)-2*ln(exp(x))+4*ln(5)))

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 44, normalized size = 1.52 \begin {gather*} 2430865342914508479353150021007861031480567253406705911367623677652226107045071656712478446533481881623815074044969719579967204481363296508789062500000000 \, x^{108} e^{\left (-4 \, x^{3} + 8 \, x^{2} \log \relax (5) + 4 \, x^{2} \log \relax (x) - 36 \, x^{2} + 72 \, x \log \relax (5) + 36 \, x \log \relax (x) - 108 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+36*x)*log(25*x/exp(x))-4*x^3-32*x^2-72*x+108)*exp((4*x^2+36*x+108)*log(25*x/exp(x))+8*log(2)
)/x,x, algorithm="maxima")

[Out]

24308653429145084793531500210078610314805672534067059113676236776522261070450716567124784465334818816238150740
44969719579967204481363296508789062500000000*x^108*e^(-4*x^3 + 8*x^2*log(5) + 4*x^2*log(x) - 36*x^2 + 72*x*log
(5) + 36*x*log(x) - 108*x)

________________________________________________________________________________________

mupad [B]  time = 3.24, size = 42, normalized size = 1.45 \begin {gather*} 256\,5^{8\,x^2+72\,x+216}\,x^{4\,x^2+36\,x+108}\,{\mathrm {e}}^{-108\,x}\,{\mathrm {e}}^{-4\,x^3}\,{\mathrm {e}}^{-36\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(8*log(2) + log(25*x*exp(-x))*(36*x + 4*x^2 + 108))*(72*x - log(25*x*exp(-x))*(36*x + 8*x^2) + 32*x^2
+ 4*x^3 - 108))/x,x)

[Out]

256*5^(72*x + 8*x^2 + 216)*x^(36*x + 4*x^2 + 108)*exp(-108*x)*exp(-4*x^3)*exp(-36*x^2)

________________________________________________________________________________________

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(((8*x**2+36*x)*ln(25*x/exp(x))-4*x**3-32*x**2-72*x+108)*exp((4*x**2+36*x+108)*ln(25*x/exp(x))+8*ln(2
))/x,x)

[Out]

Timed out

________________________________________________________________________________________