Optimal. Leaf size=50 \[ -\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-27 \left (9-e^{2 x}\right )^{3/2} \]
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Rubi [A] time = 0.0362776, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2248, 43} \[ -\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-27 \left (9-e^{2 x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int e^{6 x} \sqrt{9-e^{2 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{9-x} x^2 \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (81 \sqrt{9-x}-18 (9-x)^{3/2}+(9-x)^{5/2}\right ) \, dx,x,e^{2 x}\right )\\ &=-27 \left (9-e^{2 x}\right )^{3/2}+\frac{18}{5} \left (9-e^{2 x}\right )^{5/2}-\frac{1}{7} \left (9-e^{2 x}\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0196505, size = 33, normalized size = 0.66 \[ -\frac{1}{35} \left (9-e^{2 x}\right )^{3/2} \left (36 e^{2 x}+5 e^{4 x}+216\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 46, normalized size = 0.9 \begin{align*} -{\frac{ \left ({{\rm e}^{x}} \right ) ^{4}}{7} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{36\, \left ({{\rm e}^{x}} \right ) ^{2}}{35} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{216}{35} \left ( 9- \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974887, size = 50, normalized size = 1. \begin{align*} -\frac{1}{7} \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{7}{2}} + \frac{18}{5} \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{5}{2}} - 27 \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.872713, size = 95, normalized size = 1.9 \begin{align*} \frac{1}{35} \,{\left (5 \, e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 108 \, e^{\left (2 \, x\right )} - 1944\right )} \sqrt{-e^{\left (2 \, x\right )} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.56376, size = 41, normalized size = 0.82 \begin{align*} \begin{cases} - \frac{\left (9 - e^{2 x}\right )^{\frac{7}{2}}}{7} + \frac{18 \left (9 - e^{2 x}\right )^{\frac{5}{2}}}{5} - 27 \left (9 - e^{2 x}\right )^{\frac{3}{2}} & \text{for}\: e^{x} < \log{\left (3 \right )} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31294, size = 72, normalized size = 1.44 \begin{align*} \frac{1}{7} \,{\left (e^{\left (2 \, x\right )} - 9\right )}^{3} \sqrt{-e^{\left (2 \, x\right )} + 9} + \frac{18}{5} \,{\left (e^{\left (2 \, x\right )} - 9\right )}^{2} \sqrt{-e^{\left (2 \, x\right )} + 9} - 27 \,{\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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