Optimal. Leaf size=203 \[ 512 a^3 e^6 x-\frac {1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {128}{3} e^5 x^9 \left (d^4-4 a e^3\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )-\frac {384}{5} a e^4 x^5 \left (d^4-4 a e^3\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )+\frac {24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]
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Rubi [A] time = 0.12, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2061} \[ -\frac {1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+512 a^3 e^6 x-\frac {128}{3} e^5 x^9 \left (d^4-4 a e^3\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )+\frac {24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )-\frac {384}{5} a e^4 x^5 \left (d^4-4 a e^3\right )+8 a d^6 e^2 x^3+\frac {1536}{11} d^2 e^7 x^{11}+32 d^3 e^6 x^{10}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]
Antiderivative was successfully verified.
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Rule 2061
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 )^3 \, dx &=\int \left (512 a^3 e^6-192 a^2 d^3 e^4 x+24 a d^6 e^2 x^2-d \left (d^8-1536 a^2 e^6\right ) x^3-384 a e^4 \left (d^4-4 a e^3\right ) x^4+24 d^3 e^2 \left (d^4-16 a e^3\right ) x^5+24 d^2 e^3 \left (d^4+64 a e^3\right ) x^6-192 d e^4 \left (d^4-16 a e^3\right ) x^7-384 e^5 \left (d^4-4 a e^3\right ) x^8+320 d^3 e^6 x^9+1536 d^2 e^7 x^{10}+1536 d e^8 x^{11}+512 e^9 x^{12}\right ) \, dx\\ &=512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac {1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4-\frac {384}{5} a e^4 \left (d^4-4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac {24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8-\frac {128}{3} e^5 \left (d^4-4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 207, normalized size = 1.02 \[ 512 a^3 e^6 x-\frac {1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3+\frac {128}{3} e^5 x^9 \left (4 a e^3-d^4\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )+\frac {384}{5} a e^4 x^5 \left (4 a e^3-d^4\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )+\frac {24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+32 d^3 e^6 x^{10}+\frac {1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac {512 e^9 x^{13}}{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 205, normalized size = 1.01 \[ \frac {512}{13} x^{13} e^{9} + 128 x^{12} e^{8} d + \frac {1536}{11} x^{11} e^{7} d^{2} + 32 x^{10} e^{6} d^{3} - \frac {128}{3} x^{9} e^{5} d^{4} + \frac {512}{3} x^{9} e^{8} a - 24 x^{8} e^{4} d^{5} + 384 x^{8} e^{7} d a + \frac {24}{7} x^{7} e^{3} d^{6} + \frac {1536}{7} x^{7} e^{6} d^{2} a + 4 x^{6} e^{2} d^{7} - 64 x^{6} e^{5} d^{3} a - \frac {384}{5} x^{5} e^{4} d^{4} a + \frac {1536}{5} x^{5} e^{7} a^{2} - \frac {1}{4} x^{4} d^{9} + 384 x^{4} e^{6} d a^{2} + 8 x^{3} e^{2} d^{6} a - 96 x^{2} e^{4} d^{3} a^{2} + 512 x e^{6} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 187, normalized size = 0.92 \[ \frac {512}{13} \, x^{13} e^{9} + 128 \, d x^{12} e^{8} + \frac {1536}{11} \, d^{2} x^{11} e^{7} + 32 \, d^{3} x^{10} e^{6} - \frac {128}{3} \, d^{4} x^{9} e^{5} - 24 \, d^{5} x^{8} e^{4} + \frac {24}{7} \, d^{6} x^{7} e^{3} + 4 \, d^{7} x^{6} e^{2} - \frac {1}{4} \, d^{9} x^{4} + \frac {512}{3} \, a x^{9} e^{8} + 384 \, a d x^{8} e^{7} + \frac {1536}{7} \, a d^{2} x^{7} e^{6} - 64 \, a d^{3} x^{6} e^{5} - \frac {384}{5} \, a d^{4} x^{5} e^{4} + 8 \, a d^{6} x^{3} e^{2} + \frac {1536}{5} \, a^{2} x^{5} e^{7} + 384 \, a^{2} d x^{4} e^{6} - 96 \, a^{2} d^{3} x^{2} e^{4} + 512 \, a^{3} x e^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 288, normalized size = 1.42 \[ \frac {512 e^{9} x^{13}}{13}+128 d \,e^{8} x^{12}+\frac {1536 d^{2} e^{7} x^{11}}{11}+32 d^{3} e^{6} x^{10}+8 a \,d^{6} e^{2} x^{3}-96 a^{2} d^{3} e^{4} x^{2}+512 a^{3} e^{6} x +\frac {\left (512 a \,e^{8}-256 d^{4} e^{5}+8 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) e^{3}\right ) x^{9}}{9}+\frac {\left (2048 a d \,e^{7}-64 d^{5} e^{4}+8 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d \,e^{2}\right ) x^{8}}{8}+\frac {\left (1536 a \,e^{6} d^{2}+24 d^{6} e^{3}\right ) x^{7}}{7}+\frac {\left (-256 a \,d^{3} e^{5}+8 d^{7} e^{2}-\left (128 a \,e^{5}-16 d^{4} e^{2}\right ) d^{3}\right ) x^{6}}{6}+\frac {\left (512 a^{2} e^{7}-256 a \,d^{4} e^{4}+8 \left (128 a \,e^{5}-16 d^{4} e^{2}\right ) a \,e^{2}\right ) x^{5}}{5}+\frac {\left (1536 a^{2} e^{6} d -d^{9}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 214, normalized size = 1.05 \[ \frac {512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac {1536}{11} \, d^{2} e^{7} x^{11} + \frac {256}{5} \, d^{3} e^{6} x^{10} - \frac {1}{4} \, d^{9} x^{4} + 512 \, a^{3} e^{6} x + \frac {4}{7} \, {\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} + \frac {96}{5} \, {\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{2} e^{4} - \frac {8}{15} \, {\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3} + \frac {8}{105} \, {\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \, {\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\right )} a e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 201, normalized size = 0.99 \[ \frac {512\,e^9\,x^{13}}{13}-x^4\,\left (\frac {d^9}{4}-384\,a^2\,d\,e^6\right )+\frac {128\,e^5\,x^9\,\left (4\,a\,e^3-d^4\right )}{3}+512\,a^3\,e^6\,x+128\,d\,e^8\,x^{12}+32\,d^3\,e^6\,x^{10}+\frac {1536\,d^2\,e^7\,x^{11}}{11}+8\,a\,d^6\,e^2\,x^3+\frac {384\,a\,e^4\,x^5\,\left (4\,a\,e^3-d^4\right )}{5}+24\,d\,e^4\,x^8\,\left (16\,a\,e^3-d^4\right )+\frac {24\,d^2\,e^3\,x^7\,\left (d^4+64\,a\,e^3\right )}{7}-96\,a^2\,d^3\,e^4\,x^2-4\,d^3\,e^2\,x^6\,\left (16\,a\,e^3-d^4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 218, normalized size = 1.07 \[ 512 a^{3} e^{6} x - 96 a^{2} d^{3} e^{4} x^{2} + 8 a d^{6} e^{2} x^{3} + 32 d^{3} e^{6} x^{10} + \frac {1536 d^{2} e^{7} x^{11}}{11} + 128 d e^{8} x^{12} + \frac {512 e^{9} x^{13}}{13} + x^{9} \left (\frac {512 a e^{8}}{3} - \frac {128 d^{4} e^{5}}{3}\right ) + x^{8} \left (384 a d e^{7} - 24 d^{5} e^{4}\right ) + x^{7} \left (\frac {1536 a d^{2} e^{6}}{7} + \frac {24 d^{6} e^{3}}{7}\right ) + x^{6} \left (- 64 a d^{3} e^{5} + 4 d^{7} e^{2}\right ) + x^{5} \left (\frac {1536 a^{2} e^{7}}{5} - \frac {384 a d^{4} e^{4}}{5}\right ) + x^{4} \left (384 a^{2} d e^{6} - \frac {d^{9}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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