Optimal. Leaf size=26 \[ e^{8 x} x \left (1-x+4 \left (16-e^4+x\right )^2\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(26)=52\).
time = 0.41, antiderivative size = 174, normalized size of antiderivative = 6.69, number of steps
used = 62, number of rules used = 3, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2227, 2225,
2207} \begin {gather*} 16 e^{8 x} x^5+1016 e^{8 x} x^4-64 e^{8 x+4} x^4+24329 e^{8 x} x^3-3056 e^{8 x+4} x^3+96 e^{8 x+8} x^3+260350 e^{8 x} x^2-48912 e^{8 x+4} x^2+3064 e^{8 x+8} x^2-64 e^{8 x+12} x^2+1050625 e^{8 x} x-262400 e^{8 x+4} x+24584 e^{8 x+8} x-1024 e^{8 x+12} x-2 e^{8 x+16}+2 e^{8 x+16} (8 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1050625 e^{8 x}+8925700 e^{8 x} x+2155787 e^{8 x} x^2+198696 e^{8 x} x^3+8208 e^{8 x} x^4+128 e^{8 x} x^5+16 e^{16+8 x} (1+8 x)-128 e^{12+8 x} \left (8+65 x+4 x^2\right )+8 e^{8+8 x} \left (3073+25350 x+3100 x^2+96 x^3\right )-16 e^{4+8 x} \left (16400+137314 x+25029 x^2+1544 x^3+32 x^4\right )\right ) \, dx\\ &=8 \int e^{8+8 x} \left (3073+25350 x+3100 x^2+96 x^3\right ) \, dx+16 \int e^{16+8 x} (1+8 x) \, dx-16 \int e^{4+8 x} \left (16400+137314 x+25029 x^2+1544 x^3+32 x^4\right ) \, dx+128 \int e^{8 x} x^5 \, dx-128 \int e^{12+8 x} \left (8+65 x+4 x^2\right ) \, dx+8208 \int e^{8 x} x^4 \, dx+198696 \int e^{8 x} x^3 \, dx+1050625 \int e^{8 x} \, dx+2155787 \int e^{8 x} x^2 \, dx+8925700 \int e^{8 x} x \, dx\\ &=\frac {1050625 e^{8 x}}{8}+\frac {2231425}{2} e^{8 x} x+\frac {2155787}{8} e^{8 x} x^2+24837 e^{8 x} x^3+1026 e^{8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+8 \int \left (3073 e^{8+8 x}+25350 e^{8+8 x} x+3100 e^{8+8 x} x^2+96 e^{8+8 x} x^3\right ) \, dx-16 \int e^{16+8 x} \, dx-16 \int \left (16400 e^{4+8 x}+137314 e^{4+8 x} x+25029 e^{4+8 x} x^2+1544 e^{4+8 x} x^3+32 e^{4+8 x} x^4\right ) \, dx-80 \int e^{8 x} x^4 \, dx-128 \int \left (8 e^{12+8 x}+65 e^{12+8 x} x+4 e^{12+8 x} x^2\right ) \, dx-4104 \int e^{8 x} x^3 \, dx-74511 \int e^{8 x} x^2 \, dx-\frac {2155787}{4} \int e^{8 x} x \, dx-\frac {2231425}{2} \int e^{8 x} \, dx\\ &=-\frac {130175 e^{8 x}}{16}-2 e^{16+8 x}+\frac {33547013}{32} e^{8 x} x+\frac {520319}{2} e^{8 x} x^2+24324 e^{8 x} x^3+1016 e^{8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+40 \int e^{8 x} x^3 \, dx-512 \int e^{12+8 x} x^2 \, dx-512 \int e^{4+8 x} x^4 \, dx+768 \int e^{8+8 x} x^3 \, dx-1024 \int e^{12+8 x} \, dx+1539 \int e^{8 x} x^2 \, dx-8320 \int e^{12+8 x} x \, dx+\frac {74511}{4} \int e^{8 x} x \, dx+24584 \int e^{8+8 x} \, dx-24704 \int e^{4+8 x} x^3 \, dx+24800 \int e^{8+8 x} x^2 \, dx+\frac {2155787}{32} \int e^{8 x} \, dx+202800 \int e^{8+8 x} x \, dx-262400 \int e^{4+8 x} \, dx-400464 \int e^{4+8 x} x^2 \, dx-2197024 \int e^{4+8 x} x \, dx\\ &=\frac {72987 e^{8 x}}{256}-32800 e^{4+8 x}+3073 e^{8+8 x}-128 e^{12+8 x}-2 e^{16+8 x}+\frac {8405381}{8} e^{8 x} x-274628 e^{4+8 x} x+25350 e^{8+8 x} x-1040 e^{12+8 x} x+\frac {2082815}{8} e^{8 x} x^2-50058 e^{4+8 x} x^2+3100 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3088 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-15 \int e^{8 x} x^2 \, dx+128 \int e^{12+8 x} x \, dx+256 \int e^{4+8 x} x^3 \, dx-288 \int e^{8+8 x} x^2 \, dx-\frac {1539}{4} \int e^{8 x} x \, dx+1040 \int e^{12+8 x} \, dx-\frac {74511}{32} \int e^{8 x} \, dx-6200 \int e^{8+8 x} x \, dx+9264 \int e^{4+8 x} x^2 \, dx-25350 \int e^{8+8 x} \, dx+100116 \int e^{4+8 x} x \, dx+274628 \int e^{4+8 x} \, dx\\ &=-\frac {381 e^{8 x}}{64}+\frac {3057}{2} e^{4+8 x}-\frac {383}{4} e^{8+8 x}+2 e^{12+8 x}-2 e^{16+8 x}+\frac {33619985}{32} e^{8 x} x-\frac {524227}{2} e^{4+8 x} x+24575 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48900 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)+\frac {15}{4} \int e^{8 x} x \, dx-16 \int e^{12+8 x} \, dx+\frac {1539}{32} \int e^{8 x} \, dx+72 \int e^{8+8 x} x \, dx-96 \int e^{4+8 x} x^2 \, dx+775 \int e^{8+8 x} \, dx-2316 \int e^{4+8 x} x \, dx-\frac {25029}{2} \int e^{4+8 x} \, dx\\ &=\frac {15 e^{8 x}}{256}-\frac {573}{16} e^{4+8 x}+\frac {9}{8} e^{8+8 x}-2 e^{16+8 x}+1050625 e^{8 x} x-262403 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-\frac {15}{32} \int e^{8 x} \, dx-9 \int e^{8+8 x} \, dx+24 \int e^{4+8 x} x \, dx+\frac {579}{2} \int e^{4+8 x} \, dx\\ &=\frac {3}{8} e^{4+8 x}-2 e^{16+8 x}+1050625 e^{8 x} x-262400 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)-3 \int e^{4+8 x} \, dx\\ &=-2 e^{16+8 x}+1050625 e^{8 x} x-262400 e^{4+8 x} x+24584 e^{8+8 x} x-1024 e^{12+8 x} x+260350 e^{8 x} x^2-48912 e^{4+8 x} x^2+3064 e^{8+8 x} x^2-64 e^{12+8 x} x^2+24329 e^{8 x} x^3-3056 e^{4+8 x} x^3+96 e^{8+8 x} x^3+1016 e^{8 x} x^4-64 e^{4+8 x} x^4+16 e^{8 x} x^5+2 e^{16+8 x} (1+8 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.38, size = 32, normalized size = 1.23 \begin {gather*} e^{8 x} x \left (1025+4 e^8+127 x+4 x^2-8 e^4 (16+x)\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(442\) vs.
\(2(26)=52\).
time = 1.24, size = 443, normalized size = 17.04 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs.
\(2 (24) = 48\).
time = 0.30, size = 368, normalized size = 14.15 \begin {gather*} \frac {1}{256} \, {\left (4096 \, x^{5} - 2560 \, x^{4} + 1280 \, x^{3} - 480 \, x^{2} + 120 \, x - 15\right )} e^{\left (8 \, x\right )} - \frac {1}{8} \, {\left (512 \, x^{4} e^{4} - 256 \, x^{3} e^{4} + 96 \, x^{2} e^{4} - 24 \, x e^{4} + 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {513}{256} \, {\left (512 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 24 \, x + 3\right )} e^{\left (8 \, x\right )} + \frac {3}{8} \, {\left (256 \, x^{3} e^{8} - 96 \, x^{2} e^{8} + 24 \, x e^{8} - 3 \, e^{8}\right )} e^{\left (8 \, x\right )} - \frac {193}{16} \, {\left (256 \, x^{3} e^{4} - 96 \, x^{2} e^{4} + 24 \, x e^{4} - 3 \, e^{4}\right )} e^{\left (8 \, x\right )} + \frac {24837}{256} \, {\left (256 \, x^{3} - 96 \, x^{2} + 24 \, x - 3\right )} e^{\left (8 \, x\right )} - 2 \, {\left (32 \, x^{2} e^{12} - 8 \, x e^{12} + e^{12}\right )} e^{\left (8 \, x\right )} + \frac {775}{8} \, {\left (32 \, x^{2} e^{8} - 8 \, x e^{8} + e^{8}\right )} e^{\left (8 \, x\right )} - \frac {25029}{16} \, {\left (32 \, x^{2} e^{4} - 8 \, x e^{4} + e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2155787}{256} \, {\left (32 \, x^{2} - 8 \, x + 1\right )} e^{\left (8 \, x\right )} + 2 \, {\left (8 \, x e^{16} - e^{16}\right )} e^{\left (8 \, x\right )} - 130 \, {\left (8 \, x e^{12} - e^{12}\right )} e^{\left (8 \, x\right )} + \frac {12675}{4} \, {\left (8 \, x e^{8} - e^{8}\right )} e^{\left (8 \, x\right )} - \frac {68657}{2} \, {\left (8 \, x e^{4} - e^{4}\right )} e^{\left (8 \, x\right )} + \frac {2231425}{16} \, {\left (8 \, x - 1\right )} e^{\left (8 \, x\right )} + \frac {1050625}{8} \, e^{\left (8 \, x\right )} + 2 \, e^{\left (8 \, x + 16\right )} - 128 \, e^{\left (8 \, x + 12\right )} + 3073 \, e^{\left (8 \, x + 8\right )} - 32800 \, e^{\left (8 \, x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (24) = 48\).
time = 0.35, size = 86, normalized size = 3.31 \begin {gather*} {\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 16 \, x e^{16} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{12} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{8} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{4} + 1050625 \, x\right )} e^{\left (8 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (20) = 40\).
time = 0.09, size = 105, normalized size = 4.04 \begin {gather*} \left (16 x^{5} - 64 x^{4} e^{4} + 1016 x^{4} - 3056 x^{3} e^{4} + 24329 x^{3} + 96 x^{3} e^{8} - 64 x^{2} e^{12} - 48912 x^{2} e^{4} + 260350 x^{2} + 3064 x^{2} e^{8} - 1024 x e^{12} - 262400 x e^{4} + 1050625 x + 24584 x e^{8} + 16 x e^{16}\right ) e^{8 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs.
\(2 (24) = 48\).
time = 0.39, size = 103, normalized size = 3.96 \begin {gather*} {\left (16 \, x^{5} + 1016 \, x^{4} + 24329 \, x^{3} + 260350 \, x^{2} + 1050625 \, x\right )} e^{\left (8 \, x\right )} + 16 \, x e^{\left (8 \, x + 16\right )} - 64 \, {\left (x^{2} + 16 \, x\right )} e^{\left (8 \, x + 12\right )} + 8 \, {\left (12 \, x^{3} + 383 \, x^{2} + 3073 \, x\right )} e^{\left (8 \, x + 8\right )} - 16 \, {\left (4 \, x^{4} + 191 \, x^{3} + 3057 \, x^{2} + 16400 \, x\right )} e^{\left (8 \, x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.36, size = 31, normalized size = 1.19 \begin {gather*} x\,{\mathrm {e}}^{8\,x}\,{\left (127\,x-128\,{\mathrm {e}}^4+4\,{\mathrm {e}}^8-8\,x\,{\mathrm {e}}^4+4\,x^2+1025\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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