3.87.12 \(\int \frac {1}{5} (7+60 e^x+10 e^{2 x}+10 x) \, dx\)

Optimal. Leaf size=20 \[ -2+e^3+\left (6+e^x\right )^2+\frac {7 x}{5}+x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2194} \begin {gather*} x^2+\frac {7 x}{5}+12 e^x+e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7 + 60*E^x + 10*E^(2*x) + 10*x)/5,x]

[Out]

12*E^x + E^(2*x) + (7*x)/5 + x^2

Rule 12

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

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 {1}{5} \int \left (7+60 e^x+10 e^{2 x}+10 x\right ) \, dx\\ &=\frac {7 x}{5}+x^2+2 \int e^{2 x} \, dx+12 \int e^x \, dx\\ &=12 e^x+e^{2 x}+\frac {7 x}{5}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} 12 e^x+e^{2 x}+\frac {7 x}{5}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7 + 60*E^x + 10*E^(2*x) + 10*x)/5,x]

[Out]

12*E^x + E^(2*x) + (7*x)/5 + x^2

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fricas [A]  time = 0.86, size = 15, normalized size = 0.75 \begin {gather*} x^{2} + \frac {7}{5} \, x + e^{\left (2 \, x\right )} + 12 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+12*exp(x)+2*x+7/5,x, algorithm="fricas")

[Out]

x^2 + 7/5*x + e^(2*x) + 12*e^x

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giac [A]  time = 0.14, size = 15, normalized size = 0.75 \begin {gather*} x^{2} + \frac {7}{5} \, x + e^{\left (2 \, x\right )} + 12 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+12*exp(x)+2*x+7/5,x, algorithm="giac")

[Out]

x^2 + 7/5*x + e^(2*x) + 12*e^x

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maple [A]  time = 0.02, size = 16, normalized size = 0.80

method	result	size
default	\(\frac {7 x}{5}+x^{2}+{\mathrm e}^{2 x}+12 \,{\mathrm e}^{x}\)	\(16\)
norman	\(\frac {7 x}{5}+x^{2}+{\mathrm e}^{2 x}+12 \,{\mathrm e}^{x}\)	\(16\)
risch	\(\frac {7 x}{5}+x^{2}+{\mathrm e}^{2 x}+12 \,{\mathrm e}^{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(x)^2+12*exp(x)+2*x+7/5,x,method=_RETURNVERBOSE)

[Out]

7/5*x+x^2+exp(x)^2+12*exp(x)

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maxima [A]  time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} x^{2} + \frac {7}{5} \, x + e^{\left (2 \, x\right )} + 12 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+12*exp(x)+2*x+7/5,x, algorithm="maxima")

[Out]

x^2 + 7/5*x + e^(2*x) + 12*e^x

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mupad [B]  time = 5.26, size = 15, normalized size = 0.75 \begin {gather*} \frac {7\,x}{5}+{\mathrm {e}}^{2\,x}+12\,{\mathrm {e}}^x+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 2*exp(2*x) + 12*exp(x) + 7/5,x)

[Out]

(7*x)/5 + exp(2*x) + 12*exp(x) + x^2

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sympy [A]  time = 0.09, size = 17, normalized size = 0.85 \begin {gather*} x^{2} + \frac {7 x}{5} + e^{2 x} + 12 e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)**2+12*exp(x)+2*x+7/5,x)

[Out]

x**2 + 7*x/5 + exp(2*x) + 12*exp(x)

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