3.34.69 \(\int \frac {1}{25} e^{e^x+\frac {1}{25} (-100+10 x^2+x^3)} (25 e^x+20 x+3 x^2) \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{25} \left (x^3+10 x^2-100\right )+e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^x + (-100 + 10*x^2 + x^3)/25)*(25*E^x + 20*x + 3*x^2))/25,x]

[Out]

E^(E^x + (-100 + 10*x^2 + x^3)/25)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{25} \int e^{e^x+\frac {1}{25} \left (-100+10 x^2+x^3\right )} \left (25 e^x+20 x+3 x^2\right ) \, dx\\ &=e^{e^x+\frac {1}{25} \left (-100+10 x^2+x^3\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 21, normalized size = 0.84 \begin {gather*} e^{-4+e^x+\frac {2 x^2}{5}+\frac {x^3}{25}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^x + (-100 + 10*x^2 + x^3)/25)*(25*E^x + 20*x + 3*x^2))/25,x]

[Out]

E^(-4 + E^x + (2*x^2)/5 + x^3/25)

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 15, normalized size = 0.60 \begin {gather*} e^{\left (\frac {1}{25} \, x^{3} + \frac {2}{5} \, x^{2} + e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(25*exp(x)+3*x^2+20*x)*exp(1/25*x^3+2/5*x^2-4)*exp(exp(x)),x, algorithm="fricas")

[Out]

e^(1/25*x^3 + 2/5*x^2 + e^x - 4)

________________________________________________________________________________________

giac [A]  time = 0.30, size = 15, normalized size = 0.60 \begin {gather*} e^{\left (\frac {1}{25} \, x^{3} + \frac {2}{5} \, x^{2} + e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(25*exp(x)+3*x^2+20*x)*exp(1/25*x^3+2/5*x^2-4)*exp(exp(x)),x, algorithm="giac")

[Out]

e^(1/25*x^3 + 2/5*x^2 + e^x - 4)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 16, normalized size = 0.64

method	result	size
risch	\({\mathrm e}^{\frac {x^{3}}{25}+\frac {2 x^{2}}{5}-4+{\mathrm e}^{x}}\)	\(16\)
norman	\({\mathrm e}^{\frac {1}{25} x^{3}+\frac {2}{5} x^{2}-4} {\mathrm e}^{{\mathrm e}^{x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(25*exp(x)+3*x^2+20*x)*exp(1/25*x^3+2/5*x^2-4)*exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

exp(1/25*x^3+2/5*x^2-4+exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 15, normalized size = 0.60 \begin {gather*} e^{\left (\frac {1}{25} \, x^{3} + \frac {2}{5} \, x^{2} + e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(25*exp(x)+3*x^2+20*x)*exp(1/25*x^3+2/5*x^2-4)*exp(exp(x)),x, algorithm="maxima")

[Out]

e^(1/25*x^3 + 2/5*x^2 + e^x - 4)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 18, normalized size = 0.72 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{\frac {2\,x^2}{5}}\,{\mathrm {e}}^{\frac {x^3}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x^2)/5 + x^3/25 - 4)*exp(exp(x))*(20*x + 25*exp(x) + 3*x^2))/25,x)

[Out]

exp(exp(x))*exp(-4)*exp((2*x^2)/5)*exp(x^3/25)

________________________________________________________________________________________

sympy [A]  time = 22.73, size = 19, normalized size = 0.76 \begin {gather*} e^{\frac {x^{3}}{25} + \frac {2 x^{2}}{5} - 4} e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(25*exp(x)+3*x**2+20*x)*exp(1/25*x**3+2/5*x**2-4)*exp(exp(x)),x)

[Out]

exp(x**3/25 + 2*x**2/5 - 4)*exp(exp(x))

________________________________________________________________________________________