Optimal. Leaf size=81 \[ \frac {2}{7} \left (9-e^x\right )^{7/2}-18 \left (9-e^x\right )^{5/2}+540 \left (9-e^x\right )^{3/2}-14580 \sqrt {9-e^x}-\frac {65610}{\sqrt {9-e^x}}+\frac {39366}{\left (9-e^x\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac {2}{7} \left (9-e^x\right )^{7/2}-18 \left (9-e^x\right )^{5/2}+540 \left (9-e^x\right )^{3/2}-14580 \sqrt {9-e^x}-\frac {65610}{\sqrt {9-e^x}}+\frac {39366}{\left (9-e^x\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{6 x}}{\left (9-e^x\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {x^5}{(9-x)^{5/2}} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {59049}{(9-x)^{5/2}}-\frac {32805}{(9-x)^{3/2}}+\frac {7290}{\sqrt {9-x}}-810 \sqrt {9-x}+45 (9-x)^{3/2}-(9-x)^{5/2}\right ) \, dx,x,e^x\right )\\ &=\frac {39366}{\left (9-e^x\right )^{3/2}}-\frac {65610}{\sqrt {9-e^x}}-14580 \sqrt {9-e^x}+540 \left (9-e^x\right )^{3/2}-18 \left (9-e^x\right )^{5/2}+\frac {2}{7} \left (9-e^x\right )^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.59 \[ -\frac {2 \left (-839808 e^x+23328 e^{2 x}+432 e^{3 x}+18 e^{4 x}+e^{5 x}+5038848\right )}{7 \left (9-e^x\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 50, normalized size = 0.62 \[ -\frac {2 \, {\left (e^{\left (5 \, x\right )} + 18 \, e^{\left (4 \, x\right )} + 432 \, e^{\left (3 \, x\right )} + 23328 \, e^{\left (2 \, x\right )} - 839808 \, e^{x} + 5038848\right )} \sqrt {-e^{x} + 9}}{7 \, {\left (e^{\left (2 \, x\right )} - 18 \, e^{x} + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 75, normalized size = 0.93 \[ -\frac {2}{7} \, {\left (e^{x} - 9\right )}^{3} \sqrt {-e^{x} + 9} - 18 \, {\left (e^{x} - 9\right )}^{2} \sqrt {-e^{x} + 9} + 540 \, {\left (-e^{x} + 9\right )}^{\frac {3}{2}} - 14580 \, \sqrt {-e^{x} + 9} - \frac {13122 \, {\left (5 \, e^{x} - 42\right )}}{{\left (e^{x} - 9\right )} \sqrt {-e^{x} + 9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 0.77 \[ \frac {39366}{\left (-{\mathrm e}^{x}+9\right )^{\frac {3}{2}}}+540 \left (-{\mathrm e}^{x}+9\right )^{\frac {3}{2}}-18 \left (-{\mathrm e}^{x}+9\right )^{\frac {5}{2}}+\frac {2 \left (-{\mathrm e}^{x}+9\right )^{\frac {7}{2}}}{7}-\frac {65610}{\sqrt {-{\mathrm e}^{x}+9}}-14580 \sqrt {-{\mathrm e}^{x}+9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 61, normalized size = 0.75 \[ \frac {2}{7} \, {\left (-e^{x} + 9\right )}^{\frac {7}{2}} - 18 \, {\left (-e^{x} + 9\right )}^{\frac {5}{2}} + 540 \, {\left (-e^{x} + 9\right )}^{\frac {3}{2}} - 14580 \, \sqrt {-e^{x} + 9} - \frac {65610}{\sqrt {-e^{x} + 9}} + \frac {39366}{{\left (-e^{x} + 9\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 38, normalized size = 0.47 \[ -\frac {2\,\left (23328\,{\mathrm {e}}^{2\,x}+432\,{\mathrm {e}}^{3\,x}+18\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}-839808\,{\mathrm {e}}^x+5038848\right )}{7\,{\left (9-{\mathrm {e}}^x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.18, size = 61, normalized size = 0.75 \[ \frac {2 \left (9 - e^{x}\right )^{\frac {7}{2}}}{7} - 18 \left (9 - e^{x}\right )^{\frac {5}{2}} + 540 \left (9 - e^{x}\right )^{\frac {3}{2}} - 14580 \sqrt {9 - e^{x}} - \frac {65610}{\sqrt {9 - e^{x}}} + \frac {39366}{\left (9 - e^{x}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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