3.16.29 \(\int \frac {e^{3+3^{\frac {1}{25+x}}-e^x-2 x} (-1250-100 x-2 x^2+e^x (-625-50 x-x^2)-3^{\frac {1}{25+x}} \log (3))}{625+50 x+x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{3+3^{\frac {1}{25+x}}-e^x-2 x} \]

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Rubi [A]  time = 1.32, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {27, 6706} \begin {gather*} e^{-2 x-e^x+3^{\frac {1}{x+25}}+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 + 3^(25 + x)^(-1) - E^x - 2*x)*(-1250 - 100*x - 2*x^2 + E^x*(-625 - 50*x - x^2) - 3^(25 + x)^(-1)*Lo
g[3]))/(625 + 50*x + x^2),x]

[Out]

E^(3 + 3^(25 + x)^(-1) - E^x - 2*x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{3+3^{\frac {1}{25+x}}-e^x-2 x} \left (-1250-100 x-2 x^2+e^x \left (-625-50 x-x^2\right )-3^{\frac {1}{25+x}} \log (3)\right )}{(25+x)^2} \, dx\\ &=e^{3+3^{\frac {1}{25+x}}-e^x-2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.29, size = 19, normalized size = 1.00 \begin {gather*} e^{3+3^{\frac {1}{25+x}}-e^x-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + 3^(25 + x)^(-1) - E^x - 2*x)*(-1250 - 100*x - 2*x^2 + E^x*(-625 - 50*x - x^2) - 3^(25 + x)^(
-1)*Log[3]))/(625 + 50*x + x^2),x]

[Out]

E^(3 + 3^(25 + x)^(-1) - E^x - 2*x)

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fricas [A]  time = 0.73, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (3^{\left (\frac {1}{x + 25}\right )} - 2 \, x - e^{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)*exp(log(3)/(x+25))+(-x^2-50*x-625)*exp(x)-2*x^2-100*x-1250)*exp(exp(log(3)/(x+25))-exp(x)+3
-2*x)/(x^2+50*x+625),x, algorithm="fricas")

[Out]

e^(3^(1/(x + 25)) - 2*x - e^x + 3)

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giac [A]  time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (3^{\left (\frac {1}{x + 25}\right )} - 2 \, x - e^{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)*exp(log(3)/(x+25))+(-x^2-50*x-625)*exp(x)-2*x^2-100*x-1250)*exp(exp(log(3)/(x+25))-exp(x)+3
-2*x)/(x^2+50*x+625),x, algorithm="giac")

[Out]

e^(3^(1/(x + 25)) - 2*x - e^x + 3)

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maple [A]  time = 0.29, size = 18, normalized size = 0.95

method	result	size
risch	\({\mathrm e}^{3^{\frac {1}{x +25}}-{\mathrm e}^{x}+3-2 x}\)	\(18\)
norman	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (3)}{x +25}}-{\mathrm e}^{x}+3-2 x}+25 \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (3)}{x +25}}-{\mathrm e}^{x}+3-2 x}}{x +25}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(3)*exp(ln(3)/(x+25))+(-x^2-50*x-625)*exp(x)-2*x^2-100*x-1250)*exp(exp(ln(3)/(x+25))-exp(x)+3-2*x)/(x^
2+50*x+625),x,method=_RETURNVERBOSE)

[Out]

exp(3^(1/(x+25))-exp(x)+3-2*x)

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maxima [A]  time = 0.66, size = 17, normalized size = 0.89 \begin {gather*} e^{\left (3^{\left (\frac {1}{x + 25}\right )} - 2 \, x - e^{x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)*exp(log(3)/(x+25))+(-x^2-50*x-625)*exp(x)-2*x^2-100*x-1250)*exp(exp(log(3)/(x+25))-exp(x)+3
-2*x)/(x^2+50*x+625),x, algorithm="maxima")

[Out]

e^(3^(1/(x + 25)) - 2*x - e^x + 3)

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mupad [B]  time = 0.22, size = 20, normalized size = 1.05 \begin {gather*} {\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^3\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^{3^{\frac {1}{x+25}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(log(3)/(x + 25)) - 2*x - exp(x) + 3)*(100*x + exp(log(3)/(x + 25))*log(3) + exp(x)*(50*x + x^2 +
625) + 2*x^2 + 1250))/(50*x + x^2 + 625),x)

[Out]

exp(-2*x)*exp(3)*exp(-exp(x))*exp(3^(1/(x + 25)))

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sympy [A]  time = 0.62, size = 17, normalized size = 0.89 \begin {gather*} e^{- 2 x - e^{x} + e^{\frac {\log {\relax (3 )}}{x + 25}} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(3)*exp(ln(3)/(x+25))+(-x**2-50*x-625)*exp(x)-2*x**2-100*x-1250)*exp(exp(ln(3)/(x+25))-exp(x)+3-
2*x)/(x**2+50*x+625),x)

[Out]

exp(-2*x - exp(x) + exp(log(3)/(x + 25)) + 3)

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