Optimal. Leaf size=16 \[ 390625 e^{-x} \left (5+\frac {2}{x^2}\right )^{16} \]
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Rubi [B] time = 10.66, antiderivative size = 168, normalized size of antiderivative = 10.50, number of steps used = 563, number of rules used = 4, integrand size = 169, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2199, 2194, 2177, 2178} \begin {gather*} \frac {25600000000 e^{-x}}{x^{32}}+\frac {1024000000000 e^{-x}}{x^{30}}+\frac {19200000000000 e^{-x}}{x^{28}}+\frac {224000000000000 e^{-x}}{x^{26}}+\frac {1820000000000000 e^{-x}}{x^{24}}+\frac {10920000000000000 e^{-x}}{x^{22}}+\frac {50050000000000000 e^{-x}}{x^{20}}+\frac {178750000000000000 e^{-x}}{x^{18}}+\frac {502734375000000000 e^{-x}}{x^{16}}+\frac {1117187500000000000 e^{-x}}{x^{14}}+\frac {1955078125000000000 e^{-x}}{x^{12}}+\frac {2666015625000000000 e^{-x}}{x^{10}}+\frac {2777099609375000000 e^{-x}}{x^8}+\frac {2136230468750000000 e^{-x}}{x^6}+\frac {1144409179687500000 e^{-x}}{x^4}+\frac {381469726562500000 e^{-x}}{x^2}+59604644775390625 e^{-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-59604644775390625 e^{-x}-\frac {819200000000 e^{-x}}{x^{33}}-\frac {25600000000 e^{-x}}{x^{32}}-\frac {30720000000000 e^{-x}}{x^{31}}-\frac {1024000000000 e^{-x}}{x^{30}}-\frac {537600000000000 e^{-x}}{x^{29}}-\frac {19200000000000 e^{-x}}{x^{28}}-\frac {5824000000000000 e^{-x}}{x^{27}}-\frac {224000000000000 e^{-x}}{x^{26}}-\frac {43680000000000000 e^{-x}}{x^{25}}-\frac {1820000000000000 e^{-x}}{x^{24}}-\frac {240240000000000000 e^{-x}}{x^{23}}-\frac {10920000000000000 e^{-x}}{x^{22}}-\frac {1001000000000000000 e^{-x}}{x^{21}}-\frac {50050000000000000 e^{-x}}{x^{20}}-\frac {3217500000000000000 e^{-x}}{x^{19}}-\frac {178750000000000000 e^{-x}}{x^{18}}-\frac {8043750000000000000 e^{-x}}{x^{17}}-\frac {502734375000000000 e^{-x}}{x^{16}}-\frac {15640625000000000000 e^{-x}}{x^{15}}-\frac {1117187500000000000 e^{-x}}{x^{14}}-\frac {23460937500000000000 e^{-x}}{x^{13}}-\frac {1955078125000000000 e^{-x}}{x^{12}}-\frac {26660156250000000000 e^{-x}}{x^{11}}-\frac {2666015625000000000 e^{-x}}{x^{10}}-\frac {22216796875000000000 e^{-x}}{x^9}-\frac {2777099609375000000 e^{-x}}{x^8}-\frac {12817382812500000000 e^{-x}}{x^7}-\frac {2136230468750000000 e^{-x}}{x^6}-\frac {4577636718750000000 e^{-x}}{x^5}-\frac {1144409179687500000 e^{-x}}{x^4}-\frac {762939453125000000 e^{-x}}{x^3}-\frac {381469726562500000 e^{-x}}{x^2}\right ) \, dx\\ &=-\left (25600000000 \int \frac {e^{-x}}{x^{32}} \, dx\right )-819200000000 \int \frac {e^{-x}}{x^{33}} \, dx-1024000000000 \int \frac {e^{-x}}{x^{30}} \, dx-19200000000000 \int \frac {e^{-x}}{x^{28}} \, dx-30720000000000 \int \frac {e^{-x}}{x^{31}} \, dx-224000000000000 \int \frac {e^{-x}}{x^{26}} \, dx-537600000000000 \int \frac {e^{-x}}{x^{29}} \, dx-1820000000000000 \int \frac {e^{-x}}{x^{24}} \, dx-5824000000000000 \int \frac {e^{-x}}{x^{27}} \, dx-10920000000000000 \int \frac {e^{-x}}{x^{22}} \, dx-43680000000000000 \int \frac {e^{-x}}{x^{25}} \, dx-50050000000000000 \int \frac {e^{-x}}{x^{20}} \, dx-59604644775390625 \int e^{-x} \, dx-178750000000000000 \int \frac {e^{-x}}{x^{18}} \, dx-240240000000000000 \int \frac {e^{-x}}{x^{23}} \, dx-381469726562500000 \int \frac {e^{-x}}{x^2} \, dx-502734375000000000 \int \frac {e^{-x}}{x^{16}} \, dx-762939453125000000 \int \frac {e^{-x}}{x^3} \, dx-1001000000000000000 \int \frac {e^{-x}}{x^{21}} \, dx-1117187500000000000 \int \frac {e^{-x}}{x^{14}} \, dx-1144409179687500000 \int \frac {e^{-x}}{x^4} \, dx-1955078125000000000 \int \frac {e^{-x}}{x^{12}} \, dx-2136230468750000000 \int \frac {e^{-x}}{x^6} \, dx-2666015625000000000 \int \frac {e^{-x}}{x^{10}} \, dx-2777099609375000000 \int \frac {e^{-x}}{x^8} \, dx-3217500000000000000 \int \frac {e^{-x}}{x^{19}} \, dx-4577636718750000000 \int \frac {e^{-x}}{x^5} \, dx-8043750000000000000 \int \frac {e^{-x}}{x^{17}} \, dx-12817382812500000000 \int \frac {e^{-x}}{x^7} \, dx-15640625000000000000 \int \frac {e^{-x}}{x^{15}} \, dx-22216796875000000000 \int \frac {e^{-x}}{x^9} \, dx-23460937500000000000 \int \frac {e^{-x}}{x^{13}} \, dx-26660156250000000000 \int \frac {e^{-x}}{x^{11}} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 19, normalized size = 1.19 \begin {gather*} \frac {390625 e^{-x} \left (2+5 x^2\right )^{16}}{x^{32}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 91, normalized size = 5.69 \begin {gather*} \frac {390625 \, {\left (152587890625 \, x^{32} + 976562500000 \, x^{30} + 2929687500000 \, x^{28} + 5468750000000 \, x^{26} + 7109375000000 \, x^{24} + 6825000000000 \, x^{22} + 5005000000000 \, x^{20} + 2860000000000 \, x^{18} + 1287000000000 \, x^{16} + 457600000000 \, x^{14} + 128128000000 \, x^{12} + 27955200000 \, x^{10} + 4659200000 \, x^{8} + 573440000 \, x^{6} + 49152000 \, x^{4} + 2621440 \, x^{2} + 65536\right )} e^{\left (-x\right )}}{x^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 156, normalized size = 9.75 \begin {gather*} \frac {390625 \, {\left (152587890625 \, x^{32} e^{\left (-x\right )} + 976562500000 \, x^{30} e^{\left (-x\right )} + 2929687500000 \, x^{28} e^{\left (-x\right )} + 5468750000000 \, x^{26} e^{\left (-x\right )} + 7109375000000 \, x^{24} e^{\left (-x\right )} + 6825000000000 \, x^{22} e^{\left (-x\right )} + 5005000000000 \, x^{20} e^{\left (-x\right )} + 2860000000000 \, x^{18} e^{\left (-x\right )} + 1287000000000 \, x^{16} e^{\left (-x\right )} + 457600000000 \, x^{14} e^{\left (-x\right )} + 128128000000 \, x^{12} e^{\left (-x\right )} + 27955200000 \, x^{10} e^{\left (-x\right )} + 4659200000 \, x^{8} e^{\left (-x\right )} + 573440000 \, x^{6} e^{\left (-x\right )} + 49152000 \, x^{4} e^{\left (-x\right )} + 2621440 \, x^{2} e^{\left (-x\right )} + 65536 \, e^{\left (-x\right )}\right )}}{x^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.67, size = 92, normalized size = 5.75
method | result | size |
gosper | \(\frac {390625 \left (152587890625 x^{32}+976562500000 x^{30}+2929687500000 x^{28}+5468750000000 x^{26}+7109375000000 x^{24}+6825000000000 x^{22}+5005000000000 x^{20}+2860000000000 x^{18}+1287000000000 x^{16}+457600000000 x^{14}+128128000000 x^{12}+27955200000 x^{10}+4659200000 x^{8}+573440000 x^{6}+49152000 x^{4}+2621440 x^{2}+65536\right ) {\mathrm e}^{-x}}{x^{32}}\) | \(92\) |
risch | \(\frac {390625 \left (152587890625 x^{32}+976562500000 x^{30}+2929687500000 x^{28}+5468750000000 x^{26}+7109375000000 x^{24}+6825000000000 x^{22}+5005000000000 x^{20}+2860000000000 x^{18}+1287000000000 x^{16}+457600000000 x^{14}+128128000000 x^{12}+27955200000 x^{10}+4659200000 x^{8}+573440000 x^{6}+49152000 x^{4}+2621440 x^{2}+65536\right ) {\mathrm e}^{-x}}{x^{32}}\) | \(92\) |
default | \(59604644775390625 \,{\mathrm e}^{-x}+\frac {1024000000000 \,{\mathrm e}^{-x}}{x^{30}}+\frac {25600000000 \,{\mathrm e}^{-x}}{x^{32}}+\frac {2136230468750000000 \,{\mathrm e}^{-x}}{x^{6}}+\frac {2777099609375000000 \,{\mathrm e}^{-x}}{x^{8}}+\frac {2666015625000000000 \,{\mathrm e}^{-x}}{x^{10}}+\frac {1955078125000000000 \,{\mathrm e}^{-x}}{x^{12}}+\frac {1117187500000000000 \,{\mathrm e}^{-x}}{x^{14}}+\frac {502734375000000000 \,{\mathrm e}^{-x}}{x^{16}}+\frac {178750000000000000 \,{\mathrm e}^{-x}}{x^{18}}+\frac {50050000000000000 \,{\mathrm e}^{-x}}{x^{20}}+\frac {10920000000000000 \,{\mathrm e}^{-x}}{x^{22}}+\frac {1820000000000000 \,{\mathrm e}^{-x}}{x^{24}}+\frac {224000000000000 \,{\mathrm e}^{-x}}{x^{26}}+\frac {19200000000000 \,{\mathrm e}^{-x}}{x^{28}}+\frac {1144409179687500000 \,{\mathrm e}^{-x}}{x^{4}}+\frac {381469726562500000 \,{\mathrm e}^{-x}}{x^{2}}\) | \(152\) |
meijerg | \(\text {Expression too large to display}\) | \(5737\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 167, normalized size = 10.44 \begin {gather*} 59604644775390625 \, e^{\left (-x\right )} + 381469726562500000 \, \Gamma \left (-1, x\right ) + 762939453125000000 \, \Gamma \left (-2, x\right ) + 1144409179687500000 \, \Gamma \left (-3, x\right ) + 4577636718750000000 \, \Gamma \left (-4, x\right ) + 2136230468750000000 \, \Gamma \left (-5, x\right ) + 12817382812500000000 \, \Gamma \left (-6, x\right ) + 2777099609375000000 \, \Gamma \left (-7, x\right ) + 22216796875000000000 \, \Gamma \left (-8, x\right ) + 2666015625000000000 \, \Gamma \left (-9, x\right ) + 26660156250000000000 \, \Gamma \left (-10, x\right ) + 1955078125000000000 \, \Gamma \left (-11, x\right ) + 23460937500000000000 \, \Gamma \left (-12, x\right ) + 1117187500000000000 \, \Gamma \left (-13, x\right ) + 15640625000000000000 \, \Gamma \left (-14, x\right ) + 502734375000000000 \, \Gamma \left (-15, x\right ) + 8043750000000000000 \, \Gamma \left (-16, x\right ) + 178750000000000000 \, \Gamma \left (-17, x\right ) + 3217500000000000000 \, \Gamma \left (-18, x\right ) + 50050000000000000 \, \Gamma \left (-19, x\right ) + 1001000000000000000 \, \Gamma \left (-20, x\right ) + 10920000000000000 \, \Gamma \left (-21, x\right ) + 240240000000000000 \, \Gamma \left (-22, x\right ) + 1820000000000000 \, \Gamma \left (-23, x\right ) + 43680000000000000 \, \Gamma \left (-24, x\right ) + 224000000000000 \, \Gamma \left (-25, x\right ) + 5824000000000000 \, \Gamma \left (-26, x\right ) + 19200000000000 \, \Gamma \left (-27, x\right ) + 537600000000000 \, \Gamma \left (-28, x\right ) + 1024000000000 \, \Gamma \left (-29, x\right ) + 30720000000000 \, \Gamma \left (-30, x\right ) + 25600000000 \, \Gamma \left (-31, x\right ) + 819200000000 \, \Gamma \left (-32, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 90, normalized size = 5.62 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,\left (59604644775390625\,x^{32}+381469726562500000\,x^{30}+1144409179687500000\,x^{28}+2136230468750000000\,x^{26}+2777099609375000000\,x^{24}+2666015625000000000\,x^{22}+1955078125000000000\,x^{20}+1117187500000000000\,x^{18}+502734375000000000\,x^{16}+178750000000000000\,x^{14}+50050000000000000\,x^{12}+10920000000000000\,x^{10}+1820000000000000\,x^8+224000000000000\,x^6+19200000000000\,x^4+1024000000000\,x^2+25600000000\right )}{x^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 88, normalized size = 5.50 \begin {gather*} \frac {\left (59604644775390625 x^{32} + 381469726562500000 x^{30} + 1144409179687500000 x^{28} + 2136230468750000000 x^{26} + 2777099609375000000 x^{24} + 2666015625000000000 x^{22} + 1955078125000000000 x^{20} + 1117187500000000000 x^{18} + 502734375000000000 x^{16} + 178750000000000000 x^{14} + 50050000000000000 x^{12} + 10920000000000000 x^{10} + 1820000000000000 x^{8} + 224000000000000 x^{6} + 19200000000000 x^{4} + 1024000000000 x^{2} + 25600000000\right ) e^{- x}}{x^{32}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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