3.16.69 \(\int \frac {e^{\frac {e^{9+2 x} (10+2 x^2)+e^{9+x} (100 x+20 x^3)+e^9 (250 x^2+50 x^4)}{x^2}} (100 e^9 x^4+e^{9+2 x} (-20+20 x+4 x^3)+e^{9+x} (-100 x+100 x^2+20 x^3+20 x^4))}{x^3} \, dx\) [1569]

Optimal. Leaf size=26 \[ e^{\frac {50 e^9 \left (\frac {e^x}{5}+x\right )^2 \left (5+x^2\right )}{x^2}} \]

[Out]

exp(10*(x^2+5)/x^2*exp(9)*(x+1/5*exp(x))*(5*x+exp(x)))

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Rubi [F]
time = 2.94, 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 {e^{9+2 x} \left (10+2 x^2\right )+e^{9+x} \left (100 x+20 x^3\right )+e^9 \left (250 x^2+50 x^4\right )}{x^2}\right ) \left (100 e^9 x^4+e^{9+2 x} \left (-20+20 x+4 x^3\right )+e^{9+x} \left (-100 x+100 x^2+20 x^3+20 x^4\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((E^(9 + 2*x)*(10 + 2*x^2) + E^(9 + x)*(100*x + 20*x^3) + E^9*(250*x^2 + 50*x^4))/x^2)*(100*E^9*x^4 + E
^(9 + 2*x)*(-20 + 20*x + 4*x^3) + E^(9 + x)*(-100*x + 100*x^2 + 20*x^3 + 20*x^4)))/x^3,x]

[Out]

20*Defer[Int][E^(9 + x + (2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2), x] + 4*Defer[Int][E^(9 + 2*x + (2*E^9*(E^x + 5*
x)^2*(5 + x^2))/x^2), x] - 20*Defer[Int][E^(9 + 2*x + (2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2)/x^3, x] - 100*Defer
[Int][E^(9 + x + (2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2)/x^2, x] + 20*Defer[Int][E^(9 + 2*x + (2*E^9*(E^x + 5*x)^
2*(5 + x^2))/x^2)/x^2, x] + 100*Defer[Int][E^(9 + x + (2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2)/x, x] + 100*Defer[I
nt][E^(9 + (2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2)*x, x] + 20*Defer[Int][E^(9 + x + (2*E^9*(E^x + 5*x)^2*(5 + x^2
))/x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} \left (100 e^9 x^4+e^{9+2 x} \left (-20+20 x+4 x^3\right )+e^{9+x} \left (-100 x+100 x^2+20 x^3+20 x^4\right )\right )}{x^3} \, dx\\ &=\int \left (100 e^{9+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x+\frac {4 \exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right ) \left (-5+5 x+x^3\right )}{x^3}+\frac {20 e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} \left (-5+5 x+x^2+x^3\right )}{x^2}\right ) \, dx\\ &=4 \int \frac {\exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right ) \left (-5+5 x+x^3\right )}{x^3} \, dx+20 \int \frac {e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} \left (-5+5 x+x^2+x^3\right )}{x^2} \, dx+100 \int e^{9+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x \, dx\\ &=4 \int \left (\exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right )-\frac {5 \exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right )}{x^3}+\frac {5 \exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right )}{x^2}\right ) \, dx+20 \int \left (e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}}-\frac {5 e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}}}{x^2}+\frac {5 e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}}}{x}+e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x\right ) \, dx+100 \int e^{9+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x \, dx\\ &=4 \int \exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right ) \, dx+20 \int e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} \, dx-20 \int \frac {\exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right )}{x^3} \, dx+20 \int \frac {\exp \left (9+2 x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}\right )}{x^2} \, dx+20 \int e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x \, dx-100 \int \frac {e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}}}{x^2} \, dx+100 \int \frac {e^{9+x+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}}}{x} \, dx+100 \int e^{9+\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 24, normalized size = 0.92 \begin {gather*} e^{\frac {2 e^9 \left (e^x+5 x\right )^2 \left (5+x^2\right )}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E^(9 + 2*x)*(10 + 2*x^2) + E^(9 + x)*(100*x + 20*x^3) + E^9*(250*x^2 + 50*x^4))/x^2)*(100*E^9*x
^4 + E^(9 + 2*x)*(-20 + 20*x + 4*x^3) + E^(9 + x)*(-100*x + 100*x^2 + 20*x^3 + 20*x^4)))/x^3,x]

[Out]

E^((2*E^9*(E^x + 5*x)^2*(5 + x^2))/x^2)

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Maple [A]
time = 0.19, size = 33, normalized size = 1.27

method	result	size
risch	\({\mathrm e}^{\frac {2 \left (x^{2}+5\right ) \left (25 x^{2} {\mathrm e}^{9}+10 x \,{\mathrm e}^{x +9}+{\mathrm e}^{2 x +9}\right )}{x^{2}}}\)	\(33\)
norman	\({\mathrm e}^{\frac {\left (2 x^{2}+10\right ) {\mathrm e}^{9} {\mathrm e}^{2 x}+\left (20 x^{3}+100 x \right ) {\mathrm e}^{9} {\mathrm e}^{x}+\left (50 x^{4}+250 x^{2}\right ) {\mathrm e}^{9}}{x^{2}}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+20*x-20)*exp(9)*exp(x)^2+(20*x^4+20*x^3+100*x^2-100*x)*exp(9)*exp(x)+100*x^4*exp(9))*exp(((2*x^2+1
0)*exp(9)*exp(x)^2+(20*x^3+100*x)*exp(9)*exp(x)+(50*x^4+250*x^2)*exp(9))/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(2*(x^2+5)*(25*x^2*exp(9)+10*x*exp(x+9)+exp(2*x+9))/x^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
time = 0.45, size = 48, normalized size = 1.85 \begin {gather*} e^{\left (50 \, x^{2} e^{9} + 20 \, x e^{\left (x + 9\right )} + \frac {100 \, e^{\left (x + 9\right )}}{x} + \frac {10 \, e^{\left (2 \, x + 9\right )}}{x^{2}} + 250 \, e^{9} + 2 \, e^{\left (2 \, x + 9\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x-20)*exp(9)*exp(x)^2+(20*x^4+20*x^3+100*x^2-100*x)*exp(9)*exp(x)+100*x^4*exp(9))*exp(((2
*x^2+10)*exp(9)*exp(x)^2+(20*x^3+100*x)*exp(9)*exp(x)+(50*x^4+250*x^2)*exp(9))/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(50*x^2*e^9 + 20*x*e^(x + 9) + 100*e^(x + 9)/x + 10*e^(2*x + 9)/x^2 + 250*e^9 + 2*e^(2*x + 9))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
time = 0.36, size = 47, normalized size = 1.81 \begin {gather*} e^{\left (\frac {2 \, {\left (25 \, {\left (x^{4} + 5 \, x^{2}\right )} e^{18} + {\left (x^{2} + 5\right )} e^{\left (2 \, x + 18\right )} + 10 \, {\left (x^{3} + 5 \, x\right )} e^{\left (x + 18\right )}\right )} e^{\left (-9\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x-20)*exp(9)*exp(x)^2+(20*x^4+20*x^3+100*x^2-100*x)*exp(9)*exp(x)+100*x^4*exp(9))*exp(((2
*x^2+10)*exp(9)*exp(x)^2+(20*x^3+100*x)*exp(9)*exp(x)+(50*x^4+250*x^2)*exp(9))/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(2*(25*(x^4 + 5*x^2)*e^18 + (x^2 + 5)*e^(2*x + 18) + 10*(x^3 + 5*x)*e^(x + 18))*e^(-9)/x^2)

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Sympy [A]
time = 0.23, size = 48, normalized size = 1.85 \begin {gather*} e^{\frac {\left (2 x^{2} + 10\right ) e^{9} e^{2 x} + \left (20 x^{3} + 100 x\right ) e^{9} e^{x} + \left (50 x^{4} + 250 x^{2}\right ) e^{9}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+20*x-20)*exp(9)*exp(x)**2+(20*x**4+20*x**3+100*x**2-100*x)*exp(9)*exp(x)+100*x**4*exp(9))*e
xp(((2*x**2+10)*exp(9)*exp(x)**2+(20*x**3+100*x)*exp(9)*exp(x)+(50*x**4+250*x**2)*exp(9))/x**2)/x**3,x)

[Out]

exp(((2*x**2 + 10)*exp(9)*exp(2*x) + (20*x**3 + 100*x)*exp(9)*exp(x) + (50*x**4 + 250*x**2)*exp(9))/x**2)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+20*x-20)*exp(9)*exp(x)^2+(20*x^4+20*x^3+100*x^2-100*x)*exp(9)*exp(x)+100*x^4*exp(9))*exp(((2
*x^2+10)*exp(9)*exp(x)^2+(20*x^3+100*x)*exp(9)*exp(x)+(50*x^4+250*x^2)*exp(9))/x^2)/x^3,x, algorithm="giac")

[Out]

integrate(4*(25*x^4*e^9 + (x^3 + 5*x - 5)*e^(2*x + 9) + 5*(x^4 + x^3 + 5*x^2 - 5*x)*e^(x + 9))*e^(2*(25*(x^4 +
5*x^2)*e^9 + (x^2 + 5)*e^(2*x + 9) + 10*(x^3 + 5*x)*e^(x + 9))/x^2)/x^3, x)

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Mupad [B]
time = 1.22, size = 53, normalized size = 2.04 \begin {gather*} {\mathrm {e}}^{50\,x^2\,{\mathrm {e}}^9}\,{\mathrm {e}}^{250\,{\mathrm {e}}^9}\,{\mathrm {e}}^{\frac {10\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^9}{x^2}}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^9}\,{\mathrm {e}}^{20\,x\,{\mathrm {e}}^9\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {100\,{\mathrm {e}}^9\,{\mathrm {e}}^x}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(9)*(250*x^2 + 50*x^4) + exp(9)*exp(x)*(100*x + 20*x^3) + exp(2*x)*exp(9)*(2*x^2 + 10))/x^2)*(100
*x^4*exp(9) + exp(2*x)*exp(9)*(20*x + 4*x^3 - 20) + exp(9)*exp(x)*(100*x^2 - 100*x + 20*x^3 + 20*x^4)))/x^3,x)

[Out]

exp(50*x^2*exp(9))*exp(250*exp(9))*exp((10*exp(2*x)*exp(9))/x^2)*exp(2*exp(2*x)*exp(9))*exp(20*x*exp(9)*exp(x)
)*exp((100*exp(9)*exp(x))/x)

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