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]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 50
Rule 63
Rule 208
Rule 2282
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 = 0.02, 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]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 110, normalized size = 2.08 \[ \left [\frac {\sqrt {a} \log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt {b e^{\left (d x + c\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-d x - c\right )}\right ) + 2 \, \sqrt {b e^{\left (d x + c\right )} + a}}{d}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} \sqrt {-a}}{a}\right ) + \sqrt {b e^{\left (d x + c\right )} + a}\right )}}{d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 44, normalized size = 0.83 \[ \frac {2 \, {\left (\frac {a \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {b e^{\left (d x + c\right )} + a}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 42, normalized size = 0.79 \[ \frac {-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,{\mathrm e}^{d x +c}+a}}{\sqrt {a}}\right )+2 \sqrt {b \,{\mathrm e}^{d x +c}+a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.38, size = 63, normalized size = 1.19 \[ \frac {\sqrt {a} \log \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} - \sqrt {a}}{\sqrt {b e^{\left (d x + c\right )} + a} + \sqrt {a}}\right )}{d} + \frac {2 \, \sqrt {b e^{\left (d x + c\right )} + a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.52, size = 43, normalized size = 0.81 \[ \frac {2\,\sqrt {a+b\,{\mathrm {e}}^{c+d\,x}}}{d}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}}{\sqrt {a}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b e^{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________