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 = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.41, size = 32, normalized size = 0.64 \[ \frac {1}{35} \, {\left (5 \, e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 108 \, e^{\left (2 \, x\right )} - 1944\right )} \sqrt {-e^{\left (2 \, x\right )} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 53, normalized size = 1.06 \[ \frac {1}{7} \, {\left (e^{\left (2 \, x\right )} - 9\right )}^{3} \sqrt {-e^{\left (2 \, x\right )} + 9} + \frac {18}{5} \, {\left (e^{\left (2 \, x\right )} - 9\right )}^{2} \sqrt {-e^{\left (2 \, x\right )} + 9} - 27 \, {\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.92 \[ -\frac {36 \left (-{\mathrm e}^{2 x}+9\right )^{\frac {3}{2}} {\mathrm e}^{2 x}}{35}-\frac {\left (-{\mathrm e}^{2 x}+9\right )^{\frac {3}{2}} {\mathrm e}^{4 x}}{7}-\frac {216 \left (-{\mathrm e}^{2 x}+9\right )^{\frac {3}{2}}}{35} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 37, normalized size = 0.74 \[ -\frac {1}{7} \, {\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac {7}{2}} + \frac {18}{5} \, {\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac {5}{2}} - 27 \, {\left (-e^{\left (2 \, x\right )} + 9\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 32, normalized size = 0.64 \[ -\sqrt {9-{\mathrm {e}}^{2\,x}}\,\left (\frac {108\,{\mathrm {e}}^{2\,x}}{35}+\frac {9\,{\mathrm {e}}^{4\,x}}{35}-\frac {{\mathrm {e}}^{6\,x}}{7}+\frac {1944}{35}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.18, size = 41, normalized size = 0.82 \[ \begin {cases} - \frac {\left (9 - e^{2 x}\right )^{\frac {7}{2}}}{7} + \frac {18 \left (9 - e^{2 x}\right )^{\frac {5}{2}}}{5} - 27 \left (9 - e^{2 x}\right )^{\frac {3}{2}} & \text {for}\: e^{x} < \log {\relax (3 )} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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