3.697 \(\int (a+b e^x)^4 \, dx\)

Optimal. Leaf size=53 \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*x

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Rubi [A]  time = 0.0264042, antiderivative size = 53, 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} \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^x)^4,x]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*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 )^4 \, dx &=\operatorname{Subst}\left (\int \frac{(a+b x)^4}{x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (4 a^3 b+\frac{a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx,x,e^x\right )\\ &=4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x}+a^4 x\\ \end{align*}

Mathematica [A]  time = 0.0165201, size = 53, normalized size = 1. \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^x)^4,x]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*x

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Maple [A]  time = 0.036, size = 48, normalized size = 0.9 \begin{align*}{\frac{{b}^{4} \left ({{\rm e}^{x}} \right ) ^{4}}{4}}+{\frac{4\,a{b}^{3} \left ({{\rm e}^{x}} \right ) ^{3}}{3}}+3\,{b}^{2}{a}^{2} \left ({{\rm e}^{x}} \right ) ^{2}+4\,{a}^{3}b{{\rm e}^{x}}+{a}^{4}\ln \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*exp(x))^4,x)

[Out]

1/4*b^4*exp(x)^4+4/3*a*b^3*exp(x)^3+3*b^2*a^2*exp(x)^2+4*a^3*b*exp(x)+a^4*ln(exp(x))

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Maxima [A]  time = 0.9777, size = 61, normalized size = 1.15 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^4,x, algorithm="maxima")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x

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Fricas [A]  time = 0.865348, size = 107, normalized size = 2.02 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^4,x, algorithm="fricas")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x

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Sympy [A]  time = 0.162058, size = 51, normalized size = 0.96 \begin{align*} a^{4} x + 4 a^{3} b e^{x} + 3 a^{2} b^{2} e^{2 x} + \frac{4 a b^{3} e^{3 x}}{3} + \frac{b^{4} e^{4 x}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))**4,x)

[Out]

a**4*x + 4*a**3*b*exp(x) + 3*a**2*b**2*exp(2*x) + 4*a*b**3*exp(3*x)/3 + b**4*exp(4*x)/4

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Giac [A]  time = 1.55905, size = 61, normalized size = 1.15 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(x))^4,x, algorithm="giac")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x