3.700 \(\int \sqrt{a+b e^{c+d x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 \sqrt{a+b e^{c+d x}}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{d} \]

[Out]

(2*Sqrt[a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/d

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Rubi [A]  time = 0.0348304, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2282, 50, 63, 208} \[ \frac{2 \sqrt{a+b e^{c+d x}}}{d}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*E^(c + d*x)],x]

[Out]

(2*Sqrt[a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/d

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0165842, size = 51, normalized size = 0.96 \[ \frac{2 \sqrt{a+b e^{c+d x}}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*E^(c + d*x)],x]

[Out]

(2*Sqrt[a + b*E^(c + d*x)] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/d

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Maple [A]  time = 0.182, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( 2\,\sqrt{a+b{{\rm e}^{dx+c}}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*exp(d*x+c))^(1/2),x)

[Out]

1/d*(2*(a+b*exp(d*x+c))^(1/2)-2*a^(1/2)*arctanh((a+b*exp(d*x+c))^(1/2)/a^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.79088, size = 278, normalized size = 5.25 \begin{align*} \left [\frac{\sqrt{a} \log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt{b e^{\left (d x + c\right )} + a} \sqrt{a} + 2 \, a\right )} e^{\left (-d x - c\right )}\right ) + 2 \, \sqrt{b e^{\left (d x + c\right )} + a}}{d}, \frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} + a} \sqrt{-a}}{a}\right ) + \sqrt{b e^{\left (d x + c\right )} + a}\right )}}{d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[(sqrt(a)*log((b*e^(d*x + c) - 2*sqrt(b*e^(d*x + c) + a)*sqrt(a) + 2*a)*e^(-d*x - c)) + 2*sqrt(b*e^(d*x + c) +
a))/d, 2*(sqrt(-a)*arctan(sqrt(b*e^(d*x + c) + a)*sqrt(-a)/a) + sqrt(b*e^(d*x + c) + a))/d]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b e^{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a + b*exp(c + d*x)), x)

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Giac [A]  time = 1.30406, size = 59, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (\frac{a \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b e^{\left (d x + c\right )} + a}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*exp(d*x+c))^(1/2),x, algorithm="giac")

[Out]

2*(a*arctan(sqrt(b*e^(d*x + c) + a)/sqrt(-a))/sqrt(-a) + sqrt(b*e^(d*x + c) + a))/d