3.695 \(\int (a+b e^x)^2 \, dx\)

Optimal. Leaf size=25 \[ a^2 x+2 a b e^x+\frac{1}{2} b^2 e^{2 x} \]

[Out]

2*a*b*E^x + (b^2*E^(2*x))/2 + a^2*x

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Rubi [A]  time = 0.0143958, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2282, 43} \[ a^2 x+2 a b e^x+\frac{1}{2} b^2 e^{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)^2,x]

[Out]

2*a*b*E^x + (b^2*E^(2*x))/2 + a^2*x

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+b e^x\right )^2 \, dx &=\operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,e^x\right )\\ &=2 a b e^x+\frac{1}{2} b^2 e^{2 x}+a^2 x\\ \end{align*}

Mathematica [A]  time = 0.0094567, size = 25, normalized size = 1. \[ a^2 x+2 a b e^x+\frac{1}{2} b^2 e^{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^x)^2,x]

[Out]

2*a*b*E^x + (b^2*E^(2*x))/2 + a^2*x

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Maple [A]  time = 0.037, size = 24, normalized size = 1. \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}{b}^{2}}{2}}+2\,ab{{\rm e}^{x}}+{a}^{2}\ln \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*exp(x))^2,x)

[Out]

1/2*exp(x)^2*b^2+2*a*b*exp(x)+a^2*ln(exp(x))

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Maxima [A]  time = 0.968326, size = 28, normalized size = 1.12 \begin{align*} a^{2} x + \frac{1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^2,x, algorithm="maxima")

[Out]

a^2*x + 1/2*b^2*e^(2*x) + 2*a*b*e^x

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Fricas [A]  time = 0.831677, size = 50, normalized size = 2. \begin{align*} a^{2} x + \frac{1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^2,x, algorithm="fricas")

[Out]

a^2*x + 1/2*b^2*e^(2*x) + 2*a*b*e^x

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Sympy [A]  time = 0.118013, size = 22, normalized size = 0.88 \begin{align*} a^{2} x + 2 a b e^{x} + \frac{b^{2} e^{2 x}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))**2,x)

[Out]

a**2*x + 2*a*b*exp(x) + b**2*exp(2*x)/2

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Giac [A]  time = 1.30014, size = 28, normalized size = 1.12 \begin{align*} a^{2} x + \frac{1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^2,x, algorithm="giac")

[Out]

a^2*x + 1/2*b^2*e^(2*x) + 2*a*b*e^x