3.34.54 \(\int \frac {e^{16/625}+e^{\frac {2 (-3-x+e^{16/625} (9+x))}{e^{16/625}}} (-2 x+e^{16/625} (1+2 x))}{e^{16/625}} \, dx\)

Optimal. Leaf size=21 \[ x+e^{18+2 x-\frac {2 (3+x)}{e^{16/625}}} x \]

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Rubi [B]  time = 0.16, antiderivative size = 112, normalized size of antiderivative = 5.33, number of steps used = 5, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {12, 2187, 2176, 2194} \begin {gather*} \frac {\exp \left (2 \left (\left (1-\frac {1}{e^{16/625}}\right ) x+3 \left (3-\frac {1}{e^{16/625}}\right )\right )+\frac {16}{625}\right )}{2 \left (1-e^{16/625}\right )}+\frac {\left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right ) \exp \left (2 \left (\left (1-\frac {1}{e^{16/625}}\right ) x+3 \left (3-\frac {1}{e^{16/625}}\right )\right )-\frac {16}{625}\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(16/625) + E^((2*(-3 - x + E^(16/625)*(9 + x)))/E^(16/625))*(-2*x + E^(16/625)*(1 + 2*x)))/E^(16/625),x
]

[Out]

E^(16/625 + 2*(3*(3 - E^(-16/625)) + (1 - E^(-16/625))*x))/(2*(1 - E^(16/625))) + x + (E^(-16/625 + 2*(3*(3 -
E^(-16/625)) + (1 - E^(-16/625))*x))*(E^(16/625) - 2*(1 - E^(16/625))*x))/(2*(1 - E^(-16/625)))

Rule 12

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2187

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )\right ) \, dx}{e^{16/625}}\\ &=x+\frac {\int e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right ) \, dx}{e^{16/625}}\\ &=x+\frac {\int \exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right ) \, dx}{e^{16/625}}\\ &=x+\frac {\exp \left (-\frac {16}{625}+2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}-\int \exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \, dx\\ &=-\frac {\exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}+x+\frac {\exp \left (-\frac {16}{625}+2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.57, size = 21, normalized size = 1.00 \begin {gather*} x+e^{2 \left (9+x-\frac {3+x}{e^{16/625}}\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(16/625) + E^((2*(-3 - x + E^(16/625)*(9 + x)))/E^(16/625))*(-2*x + E^(16/625)*(1 + 2*x)))/E^(16/
625),x]

[Out]

x + E^(2*(9 + x - (3 + x)/E^(16/625)))*x

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fricas [A]  time = 0.47, size = 20, normalized size = 0.95 \begin {gather*} x e^{\left (2 \, {\left ({\left (x + 9\right )} e^{\frac {16}{625}} - x - 3\right )} e^{\left (-\frac {16}{625}\right )}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+1)*exp(16/625)-2*x)*exp(((x+9)*exp(16/625)-3-x)/exp(16/625))^2+exp(16/625))/exp(16/625),x, al
gorithm="fricas")

[Out]

x*e^(2*((x + 9)*e^(16/625) - x - 3)*e^(-16/625)) + x

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giac [B]  time = 0.34, size = 88, normalized size = 4.19 \begin {gather*} \frac {1}{2} \, {\left (2 \, x e^{\frac {16}{625}} + \frac {{\left (2 \, x e^{\left (-\frac {16}{625}\right )} - 2 \, x + e^{\left (-\frac {16}{625}\right )}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + \frac {11266}{625}\right )}}{2 \, e^{\left (-\frac {16}{625}\right )} - e^{\left (-\frac {32}{625}\right )} - 1} - \frac {{\left (2 \, x e^{\left (-\frac {16}{625}\right )} - 2 \, x + 1\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + 18\right )}}{2 \, e^{\left (-\frac {16}{625}\right )} - e^{\left (-\frac {32}{625}\right )} - 1}\right )} e^{\left (-\frac {16}{625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+1)*exp(16/625)-2*x)*exp(((x+9)*exp(16/625)-3-x)/exp(16/625))^2+exp(16/625))/exp(16/625),x, al
gorithm="giac")

[Out]

1/2*(2*x*e^(16/625) + (2*x*e^(-16/625) - 2*x + e^(-16/625))*e^(-2*x*e^(-16/625) + 2*x - 6*e^(-16/625) + 11266/
625)/(2*e^(-16/625) - e^(-32/625) - 1) - (2*x*e^(-16/625) - 2*x + 1)*e^(-2*x*e^(-16/625) + 2*x - 6*e^(-16/625)
+ 18)/(2*e^(-16/625) - e^(-32/625) - 1))*e^(-16/625)

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maple [A]  time = 0.06, size = 23, normalized size = 1.10

method	result	size
risch	\(x +x \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}} x +9 \,{\mathrm e}^{\frac {16}{625}}-x -3\right ) {\mathrm e}^{-\frac {16}{625}}}\)	\(23\)
norman	\(x +x \,{\mathrm e}^{2 \left (\left (x +9\right ) {\mathrm e}^{\frac {16}{625}}-3-x \right ) {\mathrm e}^{-\frac {16}{625}}}\)	\(24\)
default	\({\mathrm e}^{-\frac {16}{625}} \left (\frac {{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{2}-\frac {3 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {12 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {9 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {32}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {2 \,{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {2 \,{\mathrm e}^{\frac {32}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+{\mathrm e}^{\frac {16}{625}} x \right )\)	\(341\)
derivativedivides	\(\frac {\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{2}-\frac {3 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {12 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {9 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {32}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {2 \,{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {2 \,{\mathrm e}^{\frac {32}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right ) {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}\)	\(355\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x+1)*exp(16/625)-2*x)*exp(((x+9)*exp(16/625)-3-x)/exp(16/625))^2+exp(16/625))/exp(16/625),x,method=_R
ETURNVERBOSE)

[Out]

x+x*exp(2*(exp(16/625)*x+9*exp(16/625)-x-3)*exp(-16/625))

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maxima [B]  time = 0.48, size = 134, normalized size = 6.38 \begin {gather*} \frac {1}{2} \, {\left (2 \, x e^{\frac {16}{625}} + \frac {{\left (2 \, x {\left (e^{\frac {11298}{625}} - e^{\frac {11282}{625}}\right )} - e^{\frac {11298}{625}}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x\right )}}{e^{\left (6 \, e^{\left (-\frac {16}{625}\right )}\right )} + e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {32}{625}\right )} - 2 \, e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {16}{625}\right )}} - \frac {{\left (2 \, x {\left (e^{\frac {11282}{625}} - e^{\frac {11266}{625}}\right )} - e^{\frac {11282}{625}}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x\right )}}{e^{\left (6 \, e^{\left (-\frac {16}{625}\right )}\right )} + e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {32}{625}\right )} - 2 \, e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {16}{625}\right )}} - \frac {e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + \frac {11266}{625}\right )}}{e^{\left (-\frac {16}{625}\right )} - 1}\right )} e^{\left (-\frac {16}{625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+1)*exp(16/625)-2*x)*exp(((x+9)*exp(16/625)-3-x)/exp(16/625))^2+exp(16/625))/exp(16/625),x, al
gorithm="maxima")

[Out]

1/2*(2*x*e^(16/625) + (2*x*(e^(11298/625) - e^(11282/625)) - e^(11298/625))*e^(-2*x*e^(-16/625) + 2*x)/(e^(6*e
^(-16/625)) + e^(6*e^(-16/625) + 32/625) - 2*e^(6*e^(-16/625) + 16/625)) - (2*x*(e^(11282/625) - e^(11266/625)
) - e^(11282/625))*e^(-2*x*e^(-16/625) + 2*x)/(e^(6*e^(-16/625)) + e^(6*e^(-16/625) + 32/625) - 2*e^(6*e^(-16/
625) + 16/625)) - e^(-2*x*e^(-16/625) + 2*x - 6*e^(-16/625) + 11266/625)/(e^(-16/625) - 1))*e^(-16/625)

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mupad [B]  time = 0.14, size = 22, normalized size = 1.05 \begin {gather*} x\,\left ({\mathrm {e}}^{-6\,{\mathrm {e}}^{-\frac {16}{625}}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{18}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-\frac {16}{625}}}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-16/625)*(exp(16/625) - exp(-2*exp(-16/625)*(x - exp(16/625)*(x + 9) + 3))*(2*x - exp(16/625)*(2*x + 1
))),x)

[Out]

x*(exp(-6*exp(-16/625))*exp(2*x)*exp(18)*exp(-2*x*exp(-16/625)) + 1)

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sympy [A]  time = 0.13, size = 24, normalized size = 1.14 \begin {gather*} x e^{\frac {2 \left (- x + \left (x + 9\right ) e^{\frac {16}{625}} - 3\right )}{e^{\frac {16}{625}}}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x+1)*exp(16/625)-2*x)*exp(((x+9)*exp(16/625)-3-x)/exp(16/625))**2+exp(16/625))/exp(16/625),x)

[Out]

x*exp(2*(-x + (x + 9)*exp(16/625) - 3)*exp(-16/625)) + x

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