Optimal. Leaf size=24 \[ x+e^{x+x \log (3)} \left (\frac {1}{2} \left (4+e^5\right )+x\right )^5 \]
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Rubi [A] time = 9.86, antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 5, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6741, 6742, 2196, 2176, 2194} \begin {gather*} \frac {1}{32} (3 e)^x \left (2 x+e^5+4\right )^5+x \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (1+\frac {e^5}{4}\right )+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx\\ &=\int \left (1+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right )\right ) \, dx\\ &=x+\frac {1}{32} \int (3 e)^x \left (4+e^5+2 x\right )^4 \left (14+e^5 (1+\log (3))+x (2+\log (9))+\log (81)\right ) \, dx\\ &=x+\frac {1}{32} \int \left (10 (3 e)^x \left (4+e^5+2 x\right )^4+(3 e)^x \left (4+e^5+2 x\right )^5 (1+\log (3))\right ) \, dx\\ &=x+\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx+\frac {1}{32} (1+\log (3)) \int (3 e)^x \left (4+e^5+2 x\right )^5 \, dx\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5 (3 e)^x \left (4+e^5+2 x\right )^4}{16 (1+\log (3))}-\frac {5}{16} \int (3 e)^x \left (4+e^5+2 x\right )^4 \, dx-\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {5 (3 e)^x \left (4+e^5+2 x\right )^3}{2 (1+\log (3))^2}+\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2}+\frac {5 \int (3 e)^x \left (4+e^5+2 x\right )^3 \, dx}{2 (1+\log (3))}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {5\ 3^{1+x} e^x \left (4+e^5+2 x\right )^2}{(1+\log (3))^3}-\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3}-\frac {15 \int (3 e)^x \left (4+e^5+2 x\right )^2 \, dx}{(1+\log (3))^2}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5-\frac {20\ 3^{1+x} e^x \left (4+e^5+2 x\right )}{(1+\log (3))^4}+\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4}+\frac {60 \int (3 e)^x \left (4+e^5+2 x\right ) \, dx}{(1+\log (3))^3}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5+\frac {40\ 3^{1+x} e^x}{(1+\log (3))^5}-\frac {120 \int (3 e)^x \, dx}{(1+\log (3))^4}\\ &=x+\frac {1}{32} (3 e)^x \left (4+e^5+2 x\right )^5\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 4.10, size = 909, normalized size = 37.88 \begin {gather*} \frac {3^x e^{25+x} (1+\log (3))^6+32\ 3^{1+x} e^x \left (5 x^4 (1+\log (3))^5+4 x^3 (1+\log (3))^4 (-5+(1+\log (3)) \log (9))-4 \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+4 x (1+\log (3)) \left (40+20 \log ^4(3)-2 \log ^2(3) (-80+\log (9))+\log (3) (130-3 \log (81))-2 \log ^3(3) (-40+\log (81))+\log (9) (1+\log (81))\right )+2 x^2 (1+\log (3))^2 \left (-20-\log (9)+\log ^2(3) (-40+\log (81))+2 \log ^3(3) \log (81)+\log (27) \log (81)-2 \log (3) (25+\log (81))\right )\right )+8\ 3^x e^{15+x} (1+\log (3))^3 \left (20+50 \log (3)+12 \log ^3(3)-\log ^2(9)+x^2 \left (5+13 \log (3)+3 \log ^3(3)+\log (9)+\log ^2(9)+\log ^2(3) (11+\log (9))\right )+\log (27) \log (81)+2 \log ^2(3) (26+\log (81))+x \left (20+56 \log (3)+12 \log ^3(3)+\log ^2(9)+\log (81)+\log (9) \log (81)+2 \log ^2(3) (24+\log (81))\right )+\log (59049)\right )+3^x e^{20+x} (1+\log (3))^4 \left (20+16 \log ^2(3)+\log (9)+\log (3) (38+\log (81))+x (1+\log (3)) (10+\log (59049))\right )+16 (1+\log (3))^3 \left ((3 e)^x x^5 (1+\log (3))^2 (2+\log (9))+16 (3 e)^x x^2 (1+\log (3))^2 (25+\log (81))+16 (3 e)^x \left (64-\log (3) (-28+\log (9))+\log (9)+\log ^2(3) (14+\log (81))\right )+4 (3 e)^x x^3 \left (50-8 \log (3) (-10+\log (9))-\log (9)+\log (27) \log (81)+2 \log ^2(3) (20+\log (81))\right )+2 x (1+\log (3)) \left (1+\log ^2(3)+\log (9)-8\ 3^{1+x} e^x \log (9)+8 (3 e)^x (-50+\log (3) \log (9)+\log (9) \log (81)+\log (6561))\right )+(3 e)^x x^4 (1+\log (3)) (-10+\log (3) (8 \log (9)+\log (81))+\log (59049))\right )+16\ 3^x e^{5+x} (1+\log (3)) \left (x^4 (1+\log (3))^4 (5+\log (243))+2 x^3 (1+\log (3))^3 \left (20+4 \log ^2(3)+\log (3) (34+6 \log (9)+\log (81))+\log (729)\right )+4 x (1+\log (3)) \left (40+8 \log ^4(3)+8 \log ^3(3) (14+\log (81))+12 \log ^2(3) (16+\log (81))+\log (3) (152+6 \log (9)+9 \log (81))+\log (6561)\right )+6 x^2 (1+\log (3))^2 \left (20+50 \log (3)+4 \log ^3(3)+\log ^2(9)+\log (27) \log (81)+2 \log ^2(3) (22+\log (6561))+\log (59049)\right )+4 \left (20+4 \log ^5(3)+4 \log ^4(3) (17+\log (81))+4 \log ^2(3) (46+3 \log (81))+\log ^3(3) (152-4 \log (9)+10 \log (81))+\log (4782969)+\log (3) (86+\log (43046721))\right )\right )+8\ 3^x e^{10+x} (1+\log (3))^2 \left (2 \left (40+16 \log ^4(3)+2 \log ^3(3) (62+3 \log (81))+\log (3) (154+9 \log (81))+\log (729)+6 \log ^2(3) (34+\log (729))\right )+x^3 (1+\log (3))^3 (10+\log (59049))+6 x (1+\log (3)) \left (20+50 \log (3)+8 \log ^3(3)+\log (27) \log (81)+3 \log ^2(3) (16+\log (81))+\log (59049)\right )+3 x^2 (1+\log (3))^2 (20+\log (3) \log (81)+(2+\log (9)) \log (6561)+\log (282429536481))\right )}{32 (1+\log (3))^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 96, normalized size = 4.00 \begin {gather*} \frac {1}{32} \, {\left (32 \, x^{5} + 320 \, x^{4} + 1280 \, x^{3} + 2560 \, x^{2} + 10 \, {\left (x + 2\right )} e^{20} + 40 \, {\left (x^{2} + 4 \, x + 4\right )} e^{15} + 80 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{10} + 80 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )} e^{5} + 2560 \, x + e^{25} + 1024\right )} e^{\left (x \log \relax (3) + x\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 240, normalized size = 10.00 \begin {gather*} x^{5} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{2} \, x^{4} e^{\left (x \log \relax (3) + x + 5\right )} + 10 \, x^{4} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{2} \, x^{3} e^{\left (x \log \relax (3) + x + 10\right )} + 20 \, x^{3} e^{\left (x \log \relax (3) + x + 5\right )} + 40 \, x^{3} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{4} \, x^{2} e^{\left (x \log \relax (3) + x + 15\right )} + 15 \, x^{2} e^{\left (x \log \relax (3) + x + 10\right )} + 60 \, x^{2} e^{\left (x \log \relax (3) + x + 5\right )} + 80 \, x^{2} e^{\left (x \log \relax (3) + x\right )} + \frac {5}{16} \, x e^{\left (x \log \relax (3) + x + 20\right )} + 5 \, x e^{\left (x \log \relax (3) + x + 15\right )} + 30 \, x e^{\left (x \log \relax (3) + x + 10\right )} + 80 \, x e^{\left (x \log \relax (3) + x + 5\right )} + 80 \, x e^{\left (x \log \relax (3) + x\right )} + x + \frac {1}{32} \, e^{\left (x \log \relax (3) + x + 25\right )} + \frac {5}{8} \, e^{\left (x \log \relax (3) + x + 20\right )} + 5 \, e^{\left (x \log \relax (3) + x + 15\right )} + 20 \, e^{\left (x \log \relax (3) + x + 10\right )} + 40 \, e^{\left (x \log \relax (3) + x + 5\right )} + 32 \, e^{\left (x \log \relax (3) + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 114, normalized size = 4.75
method | result | size |
risch | \(x +\left (32+80 x +40 \,{\mathrm e}^{5}+x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+\frac {5 x^{4} {\mathrm e}^{5}}{2}+\frac {5 x \,{\mathrm e}^{20}}{16}+\frac {5 x^{2} {\mathrm e}^{15}}{4}+\frac {5 x^{3} {\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{10}+30 x \,{\mathrm e}^{10}+5 x \,{\mathrm e}^{15}+15 x^{2} {\mathrm e}^{10}+20 x^{3} {\mathrm e}^{5}+60 x^{2} {\mathrm e}^{5}+80 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{15}+\frac {5 \,{\mathrm e}^{20}}{8}+\frac {{\mathrm e}^{25}}{32}\right ) 3^{x} {\mathrm e}^{x}\) | \(114\) |
norman | \(x +\left (\frac {{\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{15}+20 \,{\mathrm e}^{10}+40 \,{\mathrm e}^{5}+32\right ) {\mathrm e}^{x \ln \relax (3)+x}+{\mathrm e}^{x \ln \relax (3)+x} x^{5}+\left (\frac {5 \,{\mathrm e}^{5}}{2}+10\right ) x^{4} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{5}+40\right ) x^{3} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{15}}{4}+15 \,{\mathrm e}^{10}+60 \,{\mathrm e}^{5}+80\right ) x^{2} {\mathrm e}^{x \ln \relax (3)+x}+\left (\frac {5 \,{\mathrm e}^{20}}{16}+5 \,{\mathrm e}^{15}+30 \,{\mathrm e}^{10}+80 \,{\mathrm e}^{5}+80\right ) x \,{\mathrm e}^{x \ln \relax (3)+x}\) | \(154\) |
derivativedivides | \(\text {Expression too large to display}\) | \(26214\) |
default | \(\text {Expression too large to display}\) | \(26214\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -64 \, e^{\left (-\frac {1}{2} \, {\left (e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right )} E_{1}\left (-\frac {1}{2} \, {\left (2 \, x + e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right ) \log \relax (3) - 224 \, e^{\left (-\frac {1}{2} \, {\left (e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right )} E_{1}\left (-\frac {1}{2} \, {\left (2 \, x + e^{5} + 4\right )} {\left (\log \relax (3) + 1\right )}\right ) - \frac {1}{2} \, {\left (e^{5} + 4\right )} \log \left (2 \, x + e^{5} + 4\right ) + \frac {1}{2} \, e^{5} \log \left (2 \, x + e^{5} + 4\right ) + x + \frac {1}{32} \, \int \frac {{\left (64 \, x^{6} {\left (\log \relax (3) + 1\right )} + 64 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{5} + 12 \, \log \relax (3) + 17\right )} x^{5} + 80 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{10} + 2 \, {\left (12 \, \log \relax (3) + 17\right )} e^{5} + 48 \, \log \relax (3) + 88\right )} x^{4} + 160 \, {\left ({\left (\log \relax (3) + 1\right )} e^{15} + {\left (12 \, \log \relax (3) + 17\right )} e^{10} + 8 \, {\left (6 \, \log \relax (3) + 11\right )} e^{5} + 64 \, \log \relax (3) + 144\right )} x^{3} + 20 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{20} + 4 \, {\left (12 \, \log \relax (3) + 17\right )} e^{15} + 48 \, {\left (6 \, \log \relax (3) + 11\right )} e^{10} + 192 \, {\left (4 \, \log \relax (3) + 9\right )} e^{5} + 768 \, \log \relax (3) + 2048\right )} x^{2} + 4 \, {\left (3 \, {\left (\log \relax (3) + 1\right )} e^{25} + 5 \, {\left (12 \, \log \relax (3) + 17\right )} e^{20} + 80 \, {\left (6 \, \log \relax (3) + 11\right )} e^{15} + 480 \, {\left (4 \, \log \relax (3) + 9\right )} e^{10} + 1280 \, {\left (3 \, \log \relax (3) + 8\right )} e^{5} + 3072 \, \log \relax (3) + 9472\right )} x + {\left (\log \relax (3) + 1\right )} e^{30} + 2 \, {\left (12 \, \log \relax (3) + 17\right )} e^{25} + 40 \, {\left (6 \, \log \relax (3) + 11\right )} e^{20} + 320 \, {\left (4 \, \log \relax (3) + 9\right )} e^{15} + 1280 \, {\left (3 \, \log \relax (3) + 8\right )} e^{10} + 512 \, {\left (12 \, \log \relax (3) + 37\right )} e^{5}\right )} e^{\left (x \log \relax (3) + x\right )}}{2 \, x + e^{5} + 4}\,{d x} + 2 \, \log \left (2 \, x + e^{5} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 100, normalized size = 4.17 \begin {gather*} x+3^x\,x^5\,{\mathrm {e}}^x+\frac {3^x\,{\mathrm {e}}^x\,\left (1280\,{\mathrm {e}}^5+640\,{\mathrm {e}}^{10}+160\,{\mathrm {e}}^{15}+20\,{\mathrm {e}}^{20}+{\mathrm {e}}^{25}+1024\right )}{32}+\frac {3^x\,x^4\,{\mathrm {e}}^x\,\left (80\,{\mathrm {e}}^5+320\right )}{32}+\frac {5\,3^x\,x^2\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^3}{4}+\frac {5\,3^x\,x^3\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^2}{2}+\frac {5\,3^x\,x\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^4}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.55, size = 138, normalized size = 5.75 \begin {gather*} x + \frac {\left (32 x^{5} + 320 x^{4} + 80 x^{4} e^{5} + 1280 x^{3} + 640 x^{3} e^{5} + 80 x^{3} e^{10} + 2560 x^{2} + 1920 x^{2} e^{5} + 480 x^{2} e^{10} + 40 x^{2} e^{15} + 2560 x + 2560 x e^{5} + 960 x e^{10} + 160 x e^{15} + 10 x e^{20} + 1024 + 1280 e^{5} + 640 e^{10} + 160 e^{15} + 20 e^{20} + e^{25}\right ) e^{x + x \log {\relax (3 )}}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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