3.57.11 \(\int (-2+e^{169 x} (2+338 x)) \, dx\)

Optimal. Leaf size=16 \[ 2+2 \left (5-x+e^{169 x} x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.69, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2176, 2194} \begin {gather*} -2 x-\frac {2 e^{169 x}}{169}+\frac {2}{169} e^{169 x} (169 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 + E^(169*x)*(2 + 338*x),x]

[Out]

(-2*E^(169*x))/169 - 2*x + (2*E^(169*x)*(1 + 169*x))/169

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 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} &=-2 x+\int e^{169 x} (2+338 x) \, dx\\ &=-2 x+\frac {2}{169} e^{169 x} (1+169 x)-2 \int e^{169 x} \, dx\\ &=-\frac {2 e^{169 x}}{169}-2 x+\frac {2}{169} e^{169 x} (1+169 x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 12, normalized size = 0.75 \begin {gather*} -2 x+2 e^{169 x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 + E^(169*x)*(2 + 338*x),x]

[Out]

-2*x + 2*E^(169*x)*x

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 11, normalized size = 0.69 \begin {gather*} 2 \, x e^{\left (169 \, x\right )} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((338*x+2)*exp(169*x)-2,x, algorithm="fricas")

[Out]

2*x*e^(169*x) - 2*x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 11, normalized size = 0.69 \begin {gather*} 2 \, x e^{\left (169 \, x\right )} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((338*x+2)*exp(169*x)-2,x, algorithm="giac")

[Out]

2*x*e^(169*x) - 2*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 12, normalized size = 0.75

method	result	size
derivativedivides	\(-2 x +2 x \,{\mathrm e}^{169 x}\)	\(12\)
default	\(-2 x +2 x \,{\mathrm e}^{169 x}\)	\(12\)
norman	\(-2 x +2 x \,{\mathrm e}^{169 x}\)	\(12\)
risch	\(-2 x +2 x \,{\mathrm e}^{169 x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((338*x+2)*exp(169*x)-2,x,method=_RETURNVERBOSE)

[Out]

-2*x+2*x*exp(169*x)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 21, normalized size = 1.31 \begin {gather*} \frac {2}{169} \, {\left (169 \, x - 1\right )} e^{\left (169 \, x\right )} - 2 \, x + \frac {2}{169} \, e^{\left (169 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((338*x+2)*exp(169*x)-2,x, algorithm="maxima")

[Out]

2/169*(169*x - 1)*e^(169*x) - 2*x + 2/169*e^(169*x)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 9, normalized size = 0.56 \begin {gather*} 2\,x\,\left ({\mathrm {e}}^{169\,x}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(169*x)*(338*x + 2) - 2,x)

[Out]

2*x*(exp(169*x) - 1)

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 10, normalized size = 0.62 \begin {gather*} 2 x e^{169 x} - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((338*x+2)*exp(169*x)-2,x)

[Out]

2*x*exp(169*x) - 2*x

________________________________________________________________________________________