3.72.27 \(\int \frac {-2048 e^{\frac {-2-x+5 x \log (3)}{x}}+7168 e^{\frac {2 (-2-x+5 x \log (3))}{x}}-10752 e^{\frac {3 (-2-x+5 x \log (3))}{x}}+8960 e^{\frac {4 (-2-x+5 x \log (3))}{x}}-4480 e^{\frac {5 (-2-x+5 x \log (3))}{x}}+1344 e^{\frac {6 (-2-x+5 x \log (3))}{x}}-224 e^{\frac {7 (-2-x+5 x \log (3))}{x}}+16 e^{\frac {8 (-2-x+5 x \log (3))}{x}}}{x^2} \, dx\)

Optimal. Leaf size=16 \[ \left (2-243 e^{-\frac {2+x}{x}}\right )^8 \]

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Rubi [B]  time = 0.19, antiderivative size = 89, normalized size of antiderivative = 5.56, number of steps used = 10, number of rules used = 2, integrand size = 156, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {14, 2209} \begin {gather*} 12157665459056928801 e^{-\frac {16}{x}-8}-800504721583995312 e^{-\frac {14}{x}-7}+23059806794600688 e^{-\frac {12}{x}-6}-379585297030464 e^{-\frac {10}{x}-5}[phone] e^{-\frac {8}{x}-4}-25713241344 e^{-\frac {6}{x}-3}[phone] e^{-\frac {4}{x}-2}-248832 e^{-\frac {2}{x}-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2048*E^((-2 - x + 5*x*Log[3])/x) + 7168*E^((2*(-2 - x + 5*x*Log[3]))/x) - 10752*E^((3*(-2 - x + 5*x*Log[
3]))/x) + 8960*E^((4*(-2 - x + 5*x*Log[3]))/x) - 4480*E^((5*(-2 - x + 5*x*Log[3]))/x) + 1344*E^((6*(-2 - x + 5
*x*Log[3]))/x) - 224*E^((7*(-2 - x + 5*x*Log[3]))/x) + 16*E^((8*(-2 - x + 5*x*Log[3]))/x))/x^2,x]

[Out]

12157665459056928801*E^(-8 - 16/x) - 800504721583995312*E^(-7 - 14/x) + 23059806794600688*E^(-6 - 12/x) - 3795
85297030464*E^(-5 - 10/x) + 3905198529120*E^(-4 - 8/x) - 25713241344*E^(-3 - 6/x) + 105815808*E^(-2 - 4/x) - 2
48832*E^(-1 - 2/x)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {194522647344910860816 e^{-8-\frac {16}{x}}}{x^2}-\frac {11207066102175934368 e^{-7-\frac {14}{x}}}{x^2}+\frac {276717681535208256 e^{-6-\frac {12}{x}}}{x^2}-\frac {3795852970304640 e^{-5-\frac {10}{x}}}{x^2}+\frac {31241588232960 e^{-4-\frac {8}{x}}}{x^2}-\frac {154279448064 e^{-3-\frac {6}{x}}}{x^2}+\frac {423263232 e^{-2-\frac {4}{x}}}{x^2}-\frac {497664 e^{-1-\frac {2}{x}}}{x^2}\right ) \, dx\\ &=-\left (497664 \int \frac {e^{-1-\frac {2}{x}}}{x^2} \, dx\right )[phone] \int \frac {e^{-2-\frac {4}{x}}}{x^2} \, dx-154279448064 \int \frac {e^{-3-\frac {6}{x}}}{x^2} \, dx+31241588232960 \int \frac {e^{-4-\frac {8}{x}}}{x^2} \, dx-3795852970304640 \int \frac {e^{-5-\frac {10}{x}}}{x^2} \, dx+276717681535208256 \int \frac {e^{-6-\frac {12}{x}}}{x^2} \, dx-11207066102175934368 \int \frac {e^{-7-\frac {14}{x}}}{x^2} \, dx+194522647344910860816 \int \frac {e^{-8-\frac {16}{x}}}{x^2} \, dx\\ &=12157665459056928801 e^{-8-\frac {16}{x}}-800504721583995312 e^{-7-\frac {14}{x}}+23059806794600688 e^{-6-\frac {12}{x}}-379585297030464 e^{-5-\frac {10}{x}}[phone] e^{-4-\frac {8}{x}}-25713241344 e^{-3-\frac {6}{x}}[phone] e^{-2-\frac {4}{x}}-248832 e^{-1-\frac {2}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 26, normalized size = 1.62 \begin {gather*} e^{-\frac {8 (2+x)}{x}} \left (243-2 e^{1+\frac {2}{x}}\right )^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2048*E^((-2 - x + 5*x*Log[3])/x) + 7168*E^((2*(-2 - x + 5*x*Log[3]))/x) - 10752*E^((3*(-2 - x + 5*
x*Log[3]))/x) + 8960*E^((4*(-2 - x + 5*x*Log[3]))/x) - 4480*E^((5*(-2 - x + 5*x*Log[3]))/x) + 1344*E^((6*(-2 -
x + 5*x*Log[3]))/x) - 224*E^((7*(-2 - x + 5*x*Log[3]))/x) + 16*E^((8*(-2 - x + 5*x*Log[3]))/x))/x^2,x]

[Out]

(243 - 2*E^(1 + 2/x))^8/E^((8*(2 + x))/x)

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fricas [B]  time = 0.84, size = 142, normalized size = 8.88 \begin {gather*} e^{\left (\frac {8 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} - 16 \, e^{\left (\frac {7 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} + 112 \, e^{\left (\frac {6 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} - 448 \, e^{\left (\frac {5 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} + 1120 \, e^{\left (\frac {4 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} - 1792 \, e^{\left (\frac {3 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} + 1792 \, e^{\left (\frac {2 \, {\left (5 \, x \log \relax (3) - x - 2\right )}}{x}\right )} - 1024 \, e^{\left (\frac {5 \, x \log \relax (3) - x - 2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp((5*x*log(3)-x-2)/x)^8-224*exp((5*x*log(3)-x-2)/x)^7+1344*exp((5*x*log(3)-x-2)/x)^6-4480*exp(
(5*x*log(3)-x-2)/x)^5+8960*exp((5*x*log(3)-x-2)/x)^4-10752*exp((5*x*log(3)-x-2)/x)^3+7168*exp((5*x*log(3)-x-2)
/x)^2-2048*exp((5*x*log(3)-x-2)/x))/x^2,x, algorithm="fricas")

[Out]

e^(8*(5*x*log(3) - x - 2)/x) - 16*e^(7*(5*x*log(3) - x - 2)/x) + 112*e^(6*(5*x*log(3) - x - 2)/x) - 448*e^(5*(
5*x*log(3) - x - 2)/x) + 1120*e^(4*(5*x*log(3) - x - 2)/x) - 1792*e^(3*(5*x*log(3) - x - 2)/x) + 1792*e^(2*(5*
x*log(3) - x - 2)/x) - 1024*e^((5*x*log(3) - x - 2)/x)

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giac [B]  time = 0.14, size = 85, normalized size = 5.31 \begin {gather*} 243 \, {\left (50031545098999707 \, e^{28} - 1024 \, e^{\left (\frac {14}{x} + 35\right )} + 435456 \, e^{\left (\frac {12}{x} + 34\right )} - 105815808 \, e^{\left (\frac {10}{x} + 33\right )} + 16070775840 \, e^{\left (\frac {8}{x} + 32\right )} - 1562079411648 \, e^{\left (\frac {6}{x} + 31\right )} + 94896324257616 \, e^{\left (\frac {4}{x} + 30\right )} - 3294258113514384 \, e^{\left (\frac {2}{x} + 29\right )}\right )} e^{\left (-\frac {16}{x} - 36\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp((5*x*log(3)-x-2)/x)^8-224*exp((5*x*log(3)-x-2)/x)^7+1344*exp((5*x*log(3)-x-2)/x)^6-4480*exp(
(5*x*log(3)-x-2)/x)^5+8960*exp((5*x*log(3)-x-2)/x)^4-10752*exp((5*x*log(3)-x-2)/x)^3+7168*exp((5*x*log(3)-x-2)
/x)^2-2048*exp((5*x*log(3)-x-2)/x))/x^2,x, algorithm="giac")

[Out]

243*(50031545098999707*e^28 - 1024*e^(14/x + 35) + 435456*e^(12/x + 34) - 105815808*e^(10/x + 33) + 1607077584
0*e^(8/x + 32) - 1562079411648*e^(6/x + 31) + 94896324257616*e^(4/x + 30) - 3294258113514384*e^(2/x + 29))*e^(
-16/x - 36)

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maple [B]  time = 0.06, size = 90, normalized size = 5.62

method	result	size
risch	\(12157665459056928801 \,{\mathrm e}^{-\frac {8 \left (2+x \right )}{x}}-800504721583995312 \,{\mathrm e}^{-\frac {7 \left (2+x \right )}{x}}+23059806794600688 \,{\mathrm e}^{-\frac {6 \left (2+x \right )}{x}}-379585297030464 \,{\mathrm e}^{-\frac {5 \left (2+x \right )}{x}}[phone] \,{\mathrm e}^{-\frac {4 \left (2+x \right )}{x}}-25713241344 \,{\mathrm e}^{-\frac {3 \left (2+x \right )}{x}}[phone] \,{\mathrm e}^{-\frac {2 \left (2+x \right )}{x}}-248832 \,{\mathrm e}^{-\frac {2+x}{x}}\)	\(90\)
derivativedivides	\(105815808 \,{\mathrm e}^{-\frac {4}{x}-2}-25713241344 \,{\mathrm e}^{-\frac {6}{x}-3}[phone] \,{\mathrm e}^{-\frac {8}{x}-4}-379585297030464 \,{\mathrm e}^{-\frac {10}{x}-5}+23059806794600688 \,{\mathrm e}^{-\frac {12}{x}-6}-800504721583995312 \,{\mathrm e}^{-\frac {14}{x}-7}+12157665459056928801 \,{\mathrm e}^{-\frac {16}{x}-8}-1024 \,{\mathrm e}^{5 \ln \relax (3)-1-\frac {2}{x}}\)	\(126\)
default	\(105815808 \,{\mathrm e}^{-\frac {4}{x}-2}-25713241344 \,{\mathrm e}^{-\frac {6}{x}-3}[phone] \,{\mathrm e}^{-\frac {8}{x}-4}-379585297030464 \,{\mathrm e}^{-\frac {10}{x}-5}+23059806794600688 \,{\mathrm e}^{-\frac {12}{x}-6}-800504721583995312 \,{\mathrm e}^{-\frac {14}{x}-7}+12157665459056928801 \,{\mathrm e}^{-\frac {16}{x}-8}-1024 \,{\mathrm e}^{5 \ln \relax (3)-1-\frac {2}{x}}\)	\(126\)
norman	\(\frac {x \,{\mathrm e}^{\frac {40 x \ln \relax (3)-8 x -16}{x}}+1792 x \,{\mathrm e}^{\frac {10 x \ln \relax (3)-2 x -4}{x}}-1792 x \,{\mathrm e}^{\frac {15 x \ln \relax (3)-3 x -6}{x}}+1120 x \,{\mathrm e}^{\frac {20 x \ln \relax (3)-4 x -8}{x}}-448 x \,{\mathrm e}^{\frac {25 x \ln \relax (3)-5 x -10}{x}}+112 x \,{\mathrm e}^{\frac {30 x \ln \relax (3)-6 x -12}{x}}-16 x \,{\mathrm e}^{\frac {35 x \ln \relax (3)-7 x -14}{x}}-1024 \,{\mathrm e}^{\frac {5 x \ln \relax (3)-x -2}{x}} x}{x}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*exp((5*x*ln(3)-x-2)/x)^8-224*exp((5*x*ln(3)-x-2)/x)^7+1344*exp((5*x*ln(3)-x-2)/x)^6-4480*exp((5*x*ln(3
)-x-2)/x)^5+8960*exp((5*x*ln(3)-x-2)/x)^4-10752*exp((5*x*ln(3)-x-2)/x)^3+7168*exp((5*x*ln(3)-x-2)/x)^2-2048*ex
p((5*x*ln(3)-x-2)/x))/x^2,x,method=_RETURNVERBOSE)

[Out]

12157665459056928801*exp(-8*(2+x)/x)-800504721583995312*exp(-7*(2+x)/x)+23059806794600688*exp(-6*(2+x)/x)-3795
85297030464*exp(-5*(2+x)/x)[phone]*exp(-4*(2+x)/x)-25713241344*exp(-3*(2+x)/x)[phone]*exp(-2*(2+x)/x
)-248832*exp(-(2+x)/x)

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maxima [B]  time = 0.36, size = 81, normalized size = 5.06 \begin {gather*} -248832 \, e^{\left (-\frac {2}{x} - 1\right )} + 105815808 \, e^{\left (-\frac {4}{x} - 2\right )} - 25713241344 \, e^{\left (-\frac {6}{x} - 3\right )} + 3905198529120 \, e^{\left (-\frac {8}{x} - 4\right )} - 379585297030464 \, e^{\left (-\frac {10}{x} - 5\right )} + 23059806794600688 \, e^{\left (-\frac {12}{x} - 6\right )} - 800504721583995312 \, e^{\left (-\frac {14}{x} - 7\right )} + 12157665459056928801 \, e^{\left (-\frac {16}{x} - 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp((5*x*log(3)-x-2)/x)^8-224*exp((5*x*log(3)-x-2)/x)^7+1344*exp((5*x*log(3)-x-2)/x)^6-4480*exp(
(5*x*log(3)-x-2)/x)^5+8960*exp((5*x*log(3)-x-2)/x)^4-10752*exp((5*x*log(3)-x-2)/x)^3+7168*exp((5*x*log(3)-x-2)
/x)^2-2048*exp((5*x*log(3)-x-2)/x))/x^2,x, algorithm="maxima")

[Out]

-248832*e^(-2/x - 1) + 105815808*e^(-4/x - 2) - 25713241344*e^(-6/x - 3) + 3905198529120*e^(-8/x - 4) - 379585
297030464*e^(-10/x - 5) + 23059806794600688*e^(-12/x - 6) - 800504721583995312*e^(-14/x - 7) + 121576654590569
28801*e^(-16/x - 8)

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mupad [B]  time = 4.68, size = 81, normalized size = 5.06 \begin {gather*} 105815808\,{\mathrm {e}}^{-\frac {4}{x}-2}-248832\,{\mathrm {e}}^{-\frac {2}{x}-1}-25713241344\,{\mathrm {e}}^{-\frac {6}{x}-3}[phone]\,{\mathrm {e}}^{-\frac {8}{x}-4}-379585297030464\,{\mathrm {e}}^{-\frac {10}{x}-5}+23059806794600688\,{\mathrm {e}}^{-\frac {12}{x}-6}-800504721583995312\,{\mathrm {e}}^{-\frac {14}{x}-7}+12157665459056928801\,{\mathrm {e}}^{-\frac {16}{x}-8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2048*exp(-(x - 5*x*log(3) + 2)/x) - 7168*exp(-(2*(x - 5*x*log(3) + 2))/x) + 10752*exp(-(3*(x - 5*x*log(3
) + 2))/x) - 8960*exp(-(4*(x - 5*x*log(3) + 2))/x) + 4480*exp(-(5*(x - 5*x*log(3) + 2))/x) - 1344*exp(-(6*(x -
5*x*log(3) + 2))/x) + 224*exp(-(7*(x - 5*x*log(3) + 2))/x) - 16*exp(-(8*(x - 5*x*log(3) + 2))/x))/x^2,x)

[Out]

105815808*exp(- 4/x - 2) - 248832*exp(- 2/x - 1) - 25713241344*exp(- 6/x - 3) + 3905198529120*exp(- 8/x - 4) -
379585297030464*exp(- 10/x - 5) + 23059806794600688*exp(- 12/x - 6) - 800504721583995312*exp(- 14/x - 7) + 12
157665459056928801*exp(- 16/x - 8)

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sympy [B]  time = 0.28, size = 131, normalized size = 8.19 \begin {gather*} e^{\frac {8 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} - 16 e^{\frac {7 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} + 112 e^{\frac {6 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} - 448 e^{\frac {5 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} + 1120 e^{\frac {4 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} - 1792 e^{\frac {3 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} + 1792 e^{\frac {2 \left (- x + 5 x \log {\relax (3 )} - 2\right )}{x}} - 1024 e^{\frac {- x + 5 x \log {\relax (3 )} - 2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*exp((5*x*ln(3)-x-2)/x)**8-224*exp((5*x*ln(3)-x-2)/x)**7+1344*exp((5*x*ln(3)-x-2)/x)**6-4480*exp(
(5*x*ln(3)-x-2)/x)**5+8960*exp((5*x*ln(3)-x-2)/x)**4-10752*exp((5*x*ln(3)-x-2)/x)**3+7168*exp((5*x*ln(3)-x-2)/
x)**2-2048*exp((5*x*ln(3)-x-2)/x))/x**2,x)

[Out]

exp(8*(-x + 5*x*log(3) - 2)/x) - 16*exp(7*(-x + 5*x*log(3) - 2)/x) + 112*exp(6*(-x + 5*x*log(3) - 2)/x) - 448*
exp(5*(-x + 5*x*log(3) - 2)/x) + 1120*exp(4*(-x + 5*x*log(3) - 2)/x) - 1792*exp(3*(-x + 5*x*log(3) - 2)/x) + 1
792*exp(2*(-x + 5*x*log(3) - 2)/x) - 1024*exp((-x + 5*x*log(3) - 2)/x)

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