3.27.70 \(\int \frac {1}{625} (1875+e^{\frac {1}{625} (6400+1120 x+49 x^2)} (1120+98 x)) \, dx\)

Optimal. Leaf size=20 \[ 3+e^{\frac {1}{25} \left (-16-\frac {7 x}{5}\right )^2}+3 x \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 2227, 2209} \begin {gather*} 3 x+e^{\frac {1}{625} (7 x+80)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1875 + E^((6400 + 1120*x + 49*x^2)/625)*(1120 + 98*x))/625,x]

[Out]

E^((80 + 7*x)^2/625) + 3*x

Rule 12

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \left (1875+e^{\frac {1}{625} \left (6400+1120 x+49 x^2\right )} (1120+98 x)\right ) \, dx\\ &=3 x+\frac {1}{625} \int e^{\frac {1}{625} \left (6400+1120 x+49 x^2\right )} (1120+98 x) \, dx\\ &=3 x+\frac {1}{625} \int e^{\frac {1}{625} (80+7 x)^2} (1120+98 x) \, dx\\ &=e^{\frac {1}{625} (80+7 x)^2}+3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.85 \begin {gather*} e^{\frac {1}{625} (80+7 x)^2}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1875 + E^((6400 + 1120*x + 49*x^2)/625)*(1120 + 98*x))/625,x]

[Out]

E^((80 + 7*x)^2/625) + 3*x

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fricas [A]  time = 0.52, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x + e^{\left (\frac {49}{625} \, x^{2} + \frac {224}{125} \, x + \frac {256}{25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(98*x+1120)*exp(49/625*x^2+224/125*x+256/25)+3,x, algorithm="fricas")

[Out]

3*x + e^(49/625*x^2 + 224/125*x + 256/25)

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giac [A]  time = 0.18, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x + e^{\left (\frac {49}{625} \, x^{2} + \frac {224}{125} \, x + \frac {256}{25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(98*x+1120)*exp(49/625*x^2+224/125*x+256/25)+3,x, algorithm="giac")

[Out]

3*x + e^(49/625*x^2 + 224/125*x + 256/25)

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maple [A]  time = 0.02, size = 15, normalized size = 0.75

method	result	size
risch	\(3 x +{\mathrm e}^{\frac {\left (7 x +80\right )^{2}}{625}}\)	\(15\)
default	\(3 x +{\mathrm e}^{\frac {49}{625} x^{2}+\frac {224}{125} x +\frac {256}{25}}\)	\(16\)
norman	\(3 x +{\mathrm e}^{\frac {49}{625} x^{2}+\frac {224}{125} x +\frac {256}{25}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*(98*x+1120)*exp(49/625*x^2+224/125*x+256/25)+3,x,method=_RETURNVERBOSE)

[Out]

3*x+exp(1/625*(7*x+80)^2)

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maxima [A]  time = 0.34, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x + e^{\left (\frac {49}{625} \, x^{2} + \frac {224}{125} \, x + \frac {256}{25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(98*x+1120)*exp(49/625*x^2+224/125*x+256/25)+3,x, algorithm="maxima")

[Out]

3*x + e^(49/625*x^2 + 224/125*x + 256/25)

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mupad [B]  time = 1.51, size = 15, normalized size = 0.75 \begin {gather*} 3\,x+{\mathrm {e}}^{\frac {49\,x^2}{625}+\frac {224\,x}{125}+\frac {256}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((224*x)/125 + (49*x^2)/625 + 256/25)*(98*x + 1120))/625 + 3,x)

[Out]

3*x + exp((224*x)/125 + (49*x^2)/625 + 256/25)

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sympy [A]  time = 0.09, size = 19, normalized size = 0.95 \begin {gather*} 3 x + e^{\frac {49 x^{2}}{625} + \frac {224 x}{125} + \frac {256}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*(98*x+1120)*exp(49/625*x**2+224/125*x+256/25)+3,x)

[Out]

3*x + exp(49*x**2/625 + 224*x/125 + 256/25)

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