3.26.87 \(\int \frac {-1250+e^{e^{e^x}+x} (-50 x^3-50 e^{e^x+x} x^3)+e^{2 (e^{e^x}+x)} (2 x^4+2 x^5+2 e^{e^x+x} x^5)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ \frac {\left (25-e^{e^{e^x}+x} x^2\right )^2}{x^2} \]

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Rubi [F]  time = 0.84, 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 {-1250+e^{e^{e^x}+x} \left (-50 x^3-50 e^{e^x+x} x^3\right )+e^{2 \left (e^{e^x}+x\right )} \left (2 x^4+2 x^5+2 e^{e^x+x} x^5\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1250 + E^(E^E^x + x)*(-50*x^3 - 50*E^(E^x + x)*x^3) + E^(2*(E^E^x + x))*(2*x^4 + 2*x^5 + 2*E^(E^x + x)*x
^5))/x^3,x]

[Out]

625/x^2 - 50*ExpIntegralEi[E^E^x] + 2*Defer[Int][E^(2*(E^E^x + x))*x, x] + 2*Defer[Int][E^(2*(E^E^x + x))*x^2,
x] + 2*Defer[Int][E^(2*E^E^x + E^x + 3*x)*x^2, x] - 50*Defer[Subst][Defer[Int][E^(E^x + x)*x, x], x, E^x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-50 e^{e^{e^x}+x}-\frac {1250}{x^3}+2 e^{2 e^{e^x}+e^x+3 x} x^2+2 e^{e^{e^x}+2 x} \left (-25 e^{e^x}+e^{e^{e^x}} x+e^{e^{e^x}} x^2\right )\right ) \, dx\\ &=\frac {625}{x^2}+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int e^{e^{e^x}+2 x} \left (-25 e^{e^x}+e^{e^{e^x}} x+e^{e^{e^x}} x^2\right ) \, dx-50 \int e^{e^{e^x}+x} \, dx\\ &=\frac {625}{x^2}+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int e^{e^{e^x}+2 x} \left (-25 e^{e^x}+e^{e^{e^x}} x (1+x)\right ) \, dx-50 \operatorname {Subst}\left (\int e^{e^x} \, dx,x,e^x\right )\\ &=\frac {625}{x^2}+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int \left (-25 e^{e^{e^x}+e^x+2 x}+e^{2 e^{e^x}+2 x} x (1+x)\right ) \, dx-50 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^{e^x}\right )\\ &=\frac {625}{x^2}-50 \text {Ei}\left (e^{e^x}\right )+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int e^{2 e^{e^x}+2 x} x (1+x) \, dx-50 \int e^{e^{e^x}+e^x+2 x} \, dx\\ &=\frac {625}{x^2}-50 \text {Ei}\left (e^{e^x}\right )+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int e^{2 \left (e^{e^x}+x\right )} x (1+x) \, dx-50 \operatorname {Subst}\left (\int e^{e^x+x} x \, dx,x,e^x\right )\\ &=\frac {625}{x^2}-50 \text {Ei}\left (e^{e^x}\right )+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx+2 \int \left (e^{2 \left (e^{e^x}+x\right )} x+e^{2 \left (e^{e^x}+x\right )} x^2\right ) \, dx-50 \operatorname {Subst}\left (\int e^{e^x+x} x \, dx,x,e^x\right )\\ &=\frac {625}{x^2}-50 \text {Ei}\left (e^{e^x}\right )+2 \int e^{2 \left (e^{e^x}+x\right )} x \, dx+2 \int e^{2 \left (e^{e^x}+x\right )} x^2 \, dx+2 \int e^{2 e^{e^x}+e^x+3 x} x^2 \, dx-50 \operatorname {Subst}\left (\int e^{e^x+x} x \, dx,x,e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 21, normalized size = 0.95 \begin {gather*} \frac {\left (-25+e^{e^{e^x}+x} x^2\right )^2}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1250 + E^(E^E^x + x)*(-50*x^3 - 50*E^(E^x + x)*x^3) + E^(2*(E^E^x + x))*(2*x^4 + 2*x^5 + 2*E^(E^x
+ x)*x^5))/x^3,x]

[Out]

(-25 + E^(E^E^x + x)*x^2)^2/x^2

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fricas [B]  time = 0.84, size = 48, normalized size = 2.18 \begin {gather*} \frac {x^{4} e^{\left (2 \, {\left (x e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - 50 \, x^{2} e^{\left ({\left (x e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} + 625}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5*exp(x)*exp(exp(x))+2*x^5+2*x^4)*exp(1/2*x+1/2*exp(exp(x)))^4+(-50*x^3*exp(x)*exp(exp(x))-50*
x^3)*exp(1/2*x+1/2*exp(exp(x)))^2-1250)/x^3,x, algorithm="fricas")

[Out]

(x^4*e^(2*(x*e^x + e^(x + e^x))*e^(-x)) - 50*x^2*e^((x*e^x + e^(x + e^x))*e^(-x)) + 625)/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (x^{5} e^{\left (x + e^{x}\right )} + x^{5} + x^{4}\right )} e^{\left (2 \, x + 2 \, e^{\left (e^{x}\right )}\right )} - 25 \, {\left (x^{3} e^{\left (x + e^{x}\right )} + x^{3}\right )} e^{\left (x + e^{\left (e^{x}\right )}\right )} - 625\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5*exp(x)*exp(exp(x))+2*x^5+2*x^4)*exp(1/2*x+1/2*exp(exp(x)))^4+(-50*x^3*exp(x)*exp(exp(x))-50*
x^3)*exp(1/2*x+1/2*exp(exp(x)))^2-1250)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 29, normalized size = 1.32

method	result	size
risch	\(\frac {625}{x^{2}}+x^{2} {\mathrm e}^{2 x +2 \,{\mathrm e}^{{\mathrm e}^{x}}}-50 \,{\mathrm e}^{x +{\mathrm e}^{{\mathrm e}^{x}}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5*exp(x)*exp(exp(x))+2*x^5+2*x^4)*exp(1/2*x+1/2*exp(exp(x)))^4+(-50*x^3*exp(x)*exp(exp(x))-50*x^3)*e
xp(1/2*x+1/2*exp(exp(x)))^2-1250)/x^3,x,method=_RETURNVERBOSE)

[Out]

625/x^2+x^2*exp(2*x+2*exp(exp(x)))-50*exp(x+exp(exp(x)))

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maxima [A]  time = 0.55, size = 28, normalized size = 1.27 \begin {gather*} x^{2} e^{\left (2 \, x + 2 \, e^{\left (e^{x}\right )}\right )} + \frac {625}{x^{2}} - 50 \, e^{\left (x + e^{\left (e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5*exp(x)*exp(exp(x))+2*x^5+2*x^4)*exp(1/2*x+1/2*exp(exp(x)))^4+(-50*x^3*exp(x)*exp(exp(x))-50*
x^3)*exp(1/2*x+1/2*exp(exp(x)))^2-1250)/x^3,x, algorithm="maxima")

[Out]

x^2*e^(2*x + 2*e^(e^x)) + 625/x^2 - 50*e^(x + e^(e^x))

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mupad [B]  time = 0.12, size = 28, normalized size = 1.27 \begin {gather*} \frac {625}{x^2}-50\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + exp(exp(x)))*(50*x^3 + 50*x^3*exp(exp(x))*exp(x)) - exp(2*x + 2*exp(exp(x)))*(2*x^4 + 2*x^5 + 2*
x^5*exp(exp(x))*exp(x)) + 1250)/x^3,x)

[Out]

625/x^2 - 50*exp(exp(exp(x)))*exp(x) + x^2*exp(2*exp(exp(x)))*exp(2*x)

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sympy [A]  time = 4.37, size = 29, normalized size = 1.32 \begin {gather*} x^{2} e^{2 x + 2 e^{e^{x}}} - 50 e^{x + e^{e^{x}}} + \frac {625}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5*exp(x)*exp(exp(x))+2*x**5+2*x**4)*exp(1/2*x+1/2*exp(exp(x)))**4+(-50*x**3*exp(x)*exp(exp(x)
)-50*x**3)*exp(1/2*x+1/2*exp(exp(x)))**2-1250)/x**3,x)

[Out]

x**2*exp(2*x + 2*exp(exp(x))) - 50*exp(x + exp(exp(x))) + 625/x**2

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