Optimal. Leaf size=295 \[ \frac {1}{24} e^4 \left (65536 a^2 e^6+20992 a d^4 e^3+601 d^8\right ) \left (\frac {d}{4 e}+x\right )^9+\frac {\left (256 a e^3+5 d^4\right )^2 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^5}{5120}+\frac {64}{13} e^8 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^{13}+\frac {x \left (256 a e^3+5 d^4\right )^4}{1048576 e^4}-\frac {72}{11} d^2 e^6 \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^{11}-\frac {9}{224} d^2 e^2 \left (256 a e^3+5 d^4\right ) \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^7-\frac {d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac {d}{4 e}+x\right )^3}{8192 e^2}-\frac {2048}{5} d^2 e^{10} \left (\frac {d}{4 e}+x\right )^{15}+\frac {4096}{17} e^{12} \left (\frac {d}{4 e}+x\right )^{17} \]
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Rubi [A] time = 0.53, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1106, 1090} \[ \frac {1}{24} e^4 \left (65536 a^2 e^6+20992 a d^4 e^3+601 d^8\right ) \left (\frac {d}{4 e}+x\right )^9+\frac {64}{13} e^8 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^{13}-\frac {72}{11} d^2 e^6 \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^{11}-\frac {9}{224} d^2 e^2 \left (256 a e^3+5 d^4\right ) \left (256 a e^3+17 d^4\right ) \left (\frac {d}{4 e}+x\right )^7+\frac {\left (256 a e^3+5 d^4\right )^2 \left (256 a e^3+59 d^4\right ) \left (\frac {d}{4 e}+x\right )^5}{5120}-\frac {d^2 \left (256 a e^3+5 d^4\right )^3 \left (\frac {d}{4 e}+x\right )^3}{8192 e^2}+\frac {x \left (256 a e^3+5 d^4\right )^4}{1048576 e^4}-\frac {2048}{5} d^2 e^{10} \left (\frac {d}{4 e}+x\right )^{15}+\frac {4096}{17} e^{12} \left (\frac {d}{4 e}+x\right )^{17} \]
Antiderivative was successfully verified.
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Rule 1090
Rule 1106
Rubi steps
\begin {align*} \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^4 \, dx &=\operatorname {Subst}\left (\int \left (\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4\right )^4 \, dx,x,\frac {d}{4 e}+x\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {\left (5 d^4+256 a e^3\right )^4}{1048576 e^4}-\frac {3 d^2 \left (5 d^4+256 a e^3\right )^3 x^2}{8192 e^2}+\frac {27}{512} d^4 \left (5 d^4+256 a e^3\right )^2 \left (1+\frac {1}{54} \left (5+\frac {256 a e^3}{d^4}\right )\right ) x^4-\frac {27}{8} d^6 e^2 \left (5 d^4+256 a e^3\right ) \left (\frac {17}{12}+\frac {64 a e^3}{3 d^4}\right ) x^6+81 d^8 e^4 \left (1+\frac {\left (5 d^4+256 a e^3\right ) \left (77 d^4+256 a e^3\right )}{216 d^8}\right ) x^8-864 d^6 e^6 \left (\frac {17}{12}+\frac {64 a e^3}{3 d^4}\right ) x^{10}+3456 d^4 e^8 \left (1+\frac {1}{54} \left (5+\frac {256 a e^3}{d^4}\right )\right ) x^{12}-6144 d^2 e^{10} x^{14}+4096 e^{12} x^{16}\right ) \, dx,x,\frac {d}{4 e}+x\right )\\ &=\frac {\left (5 d^4+256 a e^3\right )^4 x}{1048576 e^4}-\frac {d^2 \left (5 d^4+256 a e^3\right )^3 \left (\frac {d}{4 e}+x\right )^3}{8192 e^2}+\frac {\left (5 d^4+256 a e^3\right )^2 \left (59 d^4+256 a e^3\right ) \left (\frac {d}{4 e}+x\right )^5}{5120}-\frac {9}{224} d^2 e^2 \left (5 d^4+256 a e^3\right ) \left (17 d^4+256 a e^3\right ) \left (\frac {d}{4 e}+x\right )^7+\frac {1}{24} e^4 \left (601 d^8+20992 a d^4 e^3+65536 a^2 e^6\right ) \left (\frac {d}{4 e}+x\right )^9-\frac {72}{11} d^2 e^6 \left (17 d^4+256 a e^3\right ) \left (\frac {d}{4 e}+x\right )^{11}+\frac {64}{13} e^8 \left (59 d^4+256 a e^3\right ) \left (\frac {d}{4 e}+x\right )^{13}-\frac {2048}{5} d^2 e^{10} \left (\frac {d}{4 e}+x\right )^{15}+\frac {4096}{17} e^{12} \left (\frac {d}{4 e}+x\right )^{17}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 345, normalized size = 1.17 \[ 4096 a^4 e^8 x-1024 a^3 d^3 e^6 x^2+8 a d e^2 x^4 \left (512 a^2 e^6-d^8\right )+128 a^2 d^6 e^4 x^3-4 d e^3 x^8 \left (-1536 a^2 e^6+192 a d^4 e^3+d^8\right )+\frac {128}{3} e^4 x^9 \left (64 a^2 e^6-32 a d^4 e^3+d^8\right )-\frac {32}{7} d^2 e^2 x^7 \left (-768 a^2 e^6-24 a d^4 e^3+d^8\right )+\frac {1}{5} x^5 \left (16384 a^3 e^9-6144 a^2 d^4 e^6+d^{12}\right )+\frac {2048}{13} e^8 x^{13} \left (8 a e^3-d^4\right )-512 d e^7 x^{12} \left (d^4-8 a e^3\right )+\frac {128}{5} d^3 e^5 x^{10} \left (40 a e^3+3 d^4\right )-128 a d^3 e^4 x^6 \left (8 a e^3-d^4\right )+\frac {128}{11} d^2 e^6 x^{11} \left (384 a e^3-13 d^4\right )+1024 d^3 e^9 x^{14}+\frac {8192}{5} d^2 e^{10} x^{15}+1024 d e^{11} x^{16}+\frac {4096 e^{12} x^{17}}{17} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 353, normalized size = 1.20 \[ \frac {4096}{17} x^{17} e^{12} + 1024 x^{16} e^{11} d + \frac {8192}{5} x^{15} e^{10} d^{2} + 1024 x^{14} e^{9} d^{3} - \frac {2048}{13} x^{13} e^{8} d^{4} + \frac {16384}{13} x^{13} e^{11} a - 512 x^{12} e^{7} d^{5} + 4096 x^{12} e^{10} d a - \frac {1664}{11} x^{11} e^{6} d^{6} + \frac {49152}{11} x^{11} e^{9} d^{2} a + \frac {384}{5} x^{10} e^{5} d^{7} + 1024 x^{10} e^{8} d^{3} a + \frac {128}{3} x^{9} e^{4} d^{8} - \frac {4096}{3} x^{9} e^{7} d^{4} a + \frac {8192}{3} x^{9} e^{10} a^{2} - 4 x^{8} e^{3} d^{9} - 768 x^{8} e^{6} d^{5} a + 6144 x^{8} e^{9} d a^{2} - \frac {32}{7} x^{7} e^{2} d^{10} + \frac {768}{7} x^{7} e^{5} d^{6} a + \frac {24576}{7} x^{7} e^{8} d^{2} a^{2} + 128 x^{6} e^{4} d^{7} a - 1024 x^{6} e^{7} d^{3} a^{2} + \frac {1}{5} x^{5} d^{12} - \frac {6144}{5} x^{5} e^{6} d^{4} a^{2} + \frac {16384}{5} x^{5} e^{9} a^{3} - 8 x^{4} e^{2} d^{9} a + 4096 x^{4} e^{8} d a^{3} + 128 x^{3} e^{4} d^{6} a^{2} - 1024 x^{2} e^{6} d^{3} a^{3} + 4096 x e^{8} a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 323, normalized size = 1.09 \[ \frac {4096}{17} \, x^{17} e^{12} + 1024 \, d x^{16} e^{11} + \frac {8192}{5} \, d^{2} x^{15} e^{10} + 1024 \, d^{3} x^{14} e^{9} - \frac {2048}{13} \, d^{4} x^{13} e^{8} - 512 \, d^{5} x^{12} e^{7} - \frac {1664}{11} \, d^{6} x^{11} e^{6} + \frac {384}{5} \, d^{7} x^{10} e^{5} + \frac {128}{3} \, d^{8} x^{9} e^{4} - 4 \, d^{9} x^{8} e^{3} - \frac {32}{7} \, d^{10} x^{7} e^{2} + \frac {1}{5} \, d^{12} x^{5} + \frac {16384}{13} \, a x^{13} e^{11} + 4096 \, a d x^{12} e^{10} + \frac {49152}{11} \, a d^{2} x^{11} e^{9} + 1024 \, a d^{3} x^{10} e^{8} - \frac {4096}{3} \, a d^{4} x^{9} e^{7} - 768 \, a d^{5} x^{8} e^{6} + \frac {768}{7} \, a d^{6} x^{7} e^{5} + 128 \, a d^{7} x^{6} e^{4} - 8 \, a d^{9} x^{4} e^{2} + \frac {8192}{3} \, a^{2} x^{9} e^{10} + 6144 \, a^{2} d x^{8} e^{9} + \frac {24576}{7} \, a^{2} d^{2} x^{7} e^{8} - 1024 \, a^{2} d^{3} x^{6} e^{7} - \frac {6144}{5} \, a^{2} d^{4} x^{5} e^{6} + 128 \, a^{2} d^{6} x^{3} e^{4} + \frac {16384}{5} \, a^{3} x^{5} e^{9} + 4096 \, a^{3} d x^{4} e^{8} - 1024 \, a^{3} d^{3} x^{2} e^{6} + 4096 \, a^{4} x e^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 500, normalized size = 1.69 \[ \frac {4096 e^{12} x^{17}}{17}+1024 d \,e^{11} x^{16}+\frac {8192 d^{2} e^{10} x^{15}}{5}+1024 d^{3} e^{9} x^{14}+\frac {128 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) e^{6} x^{13}}{13}+128 a^{2} d^{6} e^{4} x^{3}-1024 a^{3} d^{3} e^{6} x^{2}+4096 a^{4} e^{8} x +\frac {\left (16384 a d \,e^{10}-2048 d^{5} e^{7}+256 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d \,e^{5}\right ) x^{12}}{12}+\frac {\left (32768 a \,d^{2} e^{9}+384 d^{6} e^{6}+128 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{2} e^{4}\right ) x^{11}}{11}+\frac {\left (14336 a \,d^{3} e^{8}+256 d^{7} e^{5}-32 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{3} e^{3}\right ) x^{10}}{10}+\frac {\left (8192 a^{2} e^{10}-8192 a \,d^{4} e^{7}+128 d^{8} e^{4}+\left (128 a \,e^{5}-16 d^{4} e^{2}\right )^{2}\right ) x^{9}}{9}+\frac {\left (16384 a^{2} d \,e^{9}-2048 a \,d^{5} e^{6}-32 d^{9} e^{3}+256 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) a d \,e^{4}\right ) x^{8}}{8}+\frac {\left (24576 a^{2} d^{2} e^{8}+512 a \,d^{6} e^{5}+2 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{6}\right ) x^{7}}{7}+\frac {\left (-2048 a^{2} d^{3} e^{7}+256 a \,d^{7} e^{4}-32 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) a \,d^{3} e^{2}\right ) x^{6}}{6}+\frac {\left (-4096 a^{2} d^{4} e^{6}+d^{12}+128 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) a^{2} e^{4}\right ) x^{5}}{5}+\frac {\left (16384 a^{3} e^{8} d -32 a \,e^{2} d^{9}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 383, normalized size = 1.30 \[ \frac {4096}{17} \, e^{12} x^{17} + 1024 \, d e^{11} x^{16} + \frac {8192}{5} \, d^{2} e^{10} x^{15} + \frac {8192}{7} \, d^{3} e^{9} x^{14} + \frac {4096}{13} \, d^{4} e^{8} x^{13} + \frac {1}{5} \, d^{12} x^{5} + 4096 \, a^{4} e^{8} x - \frac {4}{7} \, {\left (7 \, e^{3} x^{8} + 8 \, d e^{2} x^{7}\right )} d^{9} + \frac {1024}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{3} e^{6} + \frac {128}{165} \, {\left (45 \, e^{6} x^{11} + 99 \, d e^{5} x^{10} + 55 \, d^{2} e^{4} x^{9}\right )} d^{6} + \frac {128}{105} \, {\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\right )} a^{2} e^{4} - \frac {512}{1001} \, {\left (286 \, e^{9} x^{14} + 924 \, d e^{8} x^{13} + 1001 \, d^{2} e^{7} x^{12} + 364 \, d^{3} e^{6} x^{11}\right )} d^{3} + \frac {8}{15015} \, {\left (2365440 \, e^{9} x^{13} + 7687680 \, d e^{8} x^{12} + 8386560 \, d^{2} e^{7} x^{11} + 3075072 \, d^{3} e^{6} x^{10} - 15015 \, d^{9} x^{4} + 34320 \, {\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} - 32032 \, {\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3}\right )} a e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 331, normalized size = 1.12 \[ x^5\,\left (\frac {16384\,a^3\,e^9}{5}-\frac {6144\,a^2\,d^4\,e^6}{5}+\frac {d^{12}}{5}\right )+x^{10}\,\left (\frac {384\,d^7\,e^5}{5}+1024\,a\,d^3\,e^8\right )-x^{11}\,\left (\frac {1664\,d^6\,e^6}{11}-\frac {49152\,a\,d^2\,e^9}{11}\right )+\frac {4096\,e^{12}\,x^{17}}{17}+\frac {2048\,e^8\,x^{13}\,\left (8\,a\,e^3-d^4\right )}{13}+\frac {128\,e^4\,x^9\,\left (64\,a^2\,e^6-32\,a\,d^4\,e^3+d^8\right )}{3}+4096\,a^4\,e^8\,x+1024\,d\,e^{11}\,x^{16}+1024\,d^3\,e^9\,x^{14}+\frac {8192\,d^2\,e^{10}\,x^{15}}{5}+512\,d\,e^7\,x^{12}\,\left (8\,a\,e^3-d^4\right )+\frac {32\,d^2\,e^2\,x^7\,\left (768\,a^2\,e^6+24\,a\,d^4\,e^3-d^8\right )}{7}-1024\,a^3\,d^3\,e^6\,x^2+128\,a^2\,d^6\,e^4\,x^3-4\,d\,e^3\,x^8\,\left (-1536\,a^2\,e^6+192\,a\,d^4\,e^3+d^8\right )-128\,a\,d^3\,e^4\,x^6\,\left (8\,a\,e^3-d^4\right )-8\,a\,d\,e^2\,x^4\,\left (d^8-512\,a^2\,e^6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 366, normalized size = 1.24 \[ 4096 a^{4} e^{8} x - 1024 a^{3} d^{3} e^{6} x^{2} + 128 a^{2} d^{6} e^{4} x^{3} + 1024 d^{3} e^{9} x^{14} + \frac {8192 d^{2} e^{10} x^{15}}{5} + 1024 d e^{11} x^{16} + \frac {4096 e^{12} x^{17}}{17} + x^{13} \left (\frac {16384 a e^{11}}{13} - \frac {2048 d^{4} e^{8}}{13}\right ) + x^{12} \left (4096 a d e^{10} - 512 d^{5} e^{7}\right ) + x^{11} \left (\frac {49152 a d^{2} e^{9}}{11} - \frac {1664 d^{6} e^{6}}{11}\right ) + x^{10} \left (1024 a d^{3} e^{8} + \frac {384 d^{7} e^{5}}{5}\right ) + x^{9} \left (\frac {8192 a^{2} e^{10}}{3} - \frac {4096 a d^{4} e^{7}}{3} + \frac {128 d^{8} e^{4}}{3}\right ) + x^{8} \left (6144 a^{2} d e^{9} - 768 a d^{5} e^{6} - 4 d^{9} e^{3}\right ) + x^{7} \left (\frac {24576 a^{2} d^{2} e^{8}}{7} + \frac {768 a d^{6} e^{5}}{7} - \frac {32 d^{10} e^{2}}{7}\right ) + x^{6} \left (- 1024 a^{2} d^{3} e^{7} + 128 a d^{7} e^{4}\right ) + x^{5} \left (\frac {16384 a^{3} e^{9}}{5} - \frac {6144 a^{2} d^{4} e^{6}}{5} + \frac {d^{12}}{5}\right ) + x^{4} \left (4096 a^{3} d e^{8} - 8 a d^{9} e^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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