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

Optimal. Leaf size=57 \[ \frac{2 \sqrt{b e^{c+d x}-a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0355086, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2282, 50, 63, 205} \[ \frac{2 \sqrt{b e^{c+d x}-a}}{d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{d} \]

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]*ArcTan[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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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} \tan ^{-1}\left (\frac{\sqrt{-a+b e^{c+d x}}}{\sqrt{a}}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0174209, size = 55, normalized size = 0.96 \[ \frac{2 \sqrt{b e^{c+d x}-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{d} \]

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

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Maple [A]  time = 0.155, size = 48, normalized size = 0.8 \begin{align*} -2\,{\frac{\sqrt{a}}{d}\arctan \left ({\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) }+2\,{\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 = 0.786176, size = 277, normalized size = 4.86 \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}}\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)) - 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 = 1.25716, size = 61, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right ) - \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*(sqrt(a)*arctan(sqrt(b*e^(d*x + c) - a)/sqrt(a)) - sqrt(b*e^(d*x + c) - a))/d