Optimal. Leaf size=210 \[ \frac {a^4 x^2}{2}+\frac {32 a^3 x^3}{3}+8 (12-a) a^2 x^4+\frac {16}{5} a \left (128-48 a+a^2\right ) x^5+\frac {2}{3} \left (1024-1536 a+192 a^2-a^3\right ) x^6-\frac {32}{7} \left (512-288 a+15 a^2\right ) x^7+8 (128-3 a) (4-a) x^8-\frac {16}{3} \left (896-128 a+a^2\right ) x^9+\frac {1}{5} \left (20480-1536 a+3 a^2\right ) x^{10}-\frac {32}{11} (928-35 a) x^{11}+\frac {8}{3} (524-9 a) x^{12}-\frac {16}{13} (464-3 a) x^{13}+\frac {2}{7} (640-a) x^{14}-\frac {224 x^{15}}{5}+8 x^{16}-\frac {16 x^{17}}{17}+\frac {x^{18}}{18} \]
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Rubi [A]
time = 0.16, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6874}
\begin {gather*} \frac {a^4 x^2}{2}+\frac {32 a^3 x^3}{3}+\frac {1}{5} \left (3 a^2-1536 a+20480\right ) x^{10}-\frac {16}{3} \left (a^2-128 a+896\right ) x^9-\frac {32}{7} \left (15 a^2-288 a+512\right ) x^7+\frac {16}{5} a \left (a^2-48 a+128\right ) x^5+8 (12-a) a^2 x^4+\frac {2}{3} \left (-a^3+192 a^2-1536 a+1024\right ) x^6+\frac {2}{7} (640-a) x^{14}-\frac {16}{13} (464-3 a) x^{13}+\frac {8}{3} (524-9 a) x^{12}-\frac {32}{11} (928-35 a) x^{11}+8 (128-3 a) (4-a) x^8+\frac {x^{18}}{18}-\frac {16 x^{17}}{17}+8 x^{16}-\frac {224 x^{15}}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 6874
Rubi steps
\begin {align*} \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx &=\int \left (a^4 x+32 a^3 x^2-32 (-12+a) a^2 x^3+16 a \left (128-48 a+a^2\right ) x^4-4 \left (-1024+1536 a-192 a^2+a^3\right ) x^5-32 \left (512-288 a+15 a^2\right ) x^6+64 (128-3 a) (4-a) x^7-48 \left (896-128 a+a^2\right ) x^8+2 \left (20480-1536 a+3 a^2\right ) x^9+32 (-928+35 a) x^{10}-32 (-524+9 a) x^{11}+16 (-464+3 a) x^{12}-4 (-640+a) x^{13}-672 x^{14}+128 x^{15}-16 x^{16}+x^{17}\right ) \, dx\\ &=\frac {a^4 x^2}{2}+\frac {32 a^3 x^3}{3}+8 (12-a) a^2 x^4+\frac {16}{5} a \left (128-48 a+a^2\right ) x^5+\frac {2}{3} \left (1024-1536 a+192 a^2-a^3\right ) x^6-\frac {32}{7} \left (512-288 a+15 a^2\right ) x^7+8 (128-3 a) (4-a) x^8-\frac {16}{3} \left (896-128 a+a^2\right ) x^9+\frac {1}{5} \left (20480-1536 a+3 a^2\right ) x^{10}-\frac {32}{11} (928-35 a) x^{11}+\frac {8}{3} (524-9 a) x^{12}-\frac {16}{13} (464-3 a) x^{13}+\frac {2}{7} (640-a) x^{14}-\frac {224 x^{15}}{5}+8 x^{16}-\frac {16 x^{17}}{17}+\frac {x^{18}}{18}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 204, normalized size = 0.97 \begin {gather*} \frac {a^4 x^2}{2}+\frac {32 a^3 x^3}{3}-8 (-12+a) a^2 x^4+\frac {16}{5} a \left (128-48 a+a^2\right ) x^5-\frac {2}{3} \left (-1024+1536 a-192 a^2+a^3\right ) x^6-\frac {32}{7} \left (512-288 a+15 a^2\right ) x^7+8 \left (512-140 a+3 a^2\right ) x^8-\frac {16}{3} \left (896-128 a+a^2\right ) x^9+\frac {1}{5} \left (20480-1536 a+3 a^2\right ) x^{10}+\frac {32}{11} (-928+35 a) x^{11}-\frac {8}{3} (-524+9 a) x^{12}+\frac {16}{13} (-464+3 a) x^{13}-\frac {2}{7} (-640+a) x^{14}-\frac {224 x^{15}}{5}+8 x^{16}-\frac {16 x^{17}}{17}+\frac {x^{18}}{18} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 267, normalized size = 1.27
method | result | size |
norman | \(\frac {a^{4} x^{2}}{2}+\frac {32 a^{3} x^{3}}{3}+\left (-8 a^{3}+96 a^{2}\right ) x^{4}+\left (\frac {16}{5} a^{3}-\frac {768}{5} a^{2}+\frac {2048}{5} a \right ) x^{5}+\left (-\frac {2}{3} a^{3}+128 a^{2}-1024 a +\frac {2048}{3}\right ) x^{6}+\left (-\frac {480}{7} a^{2}+\frac {9216}{7} a -\frac {16384}{7}\right ) x^{7}+\left (24 a^{2}-1120 a +4096\right ) x^{8}+\left (-\frac {16}{3} a^{2}+\frac {2048}{3} a -\frac {14336}{3}\right ) x^{9}+\left (\frac {3}{5} a^{2}-\frac {1536}{5} a +4096\right ) x^{10}+\left (\frac {1120 a}{11}-\frac {29696}{11}\right ) x^{11}+\left (-24 a +\frac {4192}{3}\right ) x^{12}+\left (\frac {48 a}{13}-\frac {7424}{13}\right ) x^{13}+\left (-\frac {2 a}{7}+\frac {1280}{7}\right ) x^{14}-\frac {224 x^{15}}{5}+8 x^{16}-\frac {16 x^{17}}{17}+\frac {x^{18}}{18}\) | \(182\) |
gosper | \(4096 x^{8}-\frac {14336}{3} x^{9}-\frac {16384}{7} x^{7}+\frac {2048}{3} x^{6}+\frac {1}{2} a^{4} x^{2}+24 a^{2} x^{8}-1120 a \,x^{8}+\frac {1120}{11} x^{11} a -24 x^{12} a +\frac {2048}{3} x^{9} a -\frac {1536}{5} x^{10} a +\frac {9216}{7} x^{7} a +\frac {16}{5} x^{5} a^{3}+\frac {48}{13} x^{13} a -\frac {2}{7} x^{14} a +\frac {3}{5} x^{10} a^{2}-\frac {16}{3} x^{9} a^{2}-\frac {2}{3} x^{6} a^{3}+96 a^{2} x^{4}+4096 x^{10}+\frac {32}{3} a^{3} x^{3}-1024 a \,x^{6}-8 a^{3} x^{4}+\frac {2048}{5} a \,x^{5}-\frac {480}{7} a^{2} x^{7}+128 a^{2} x^{6}-\frac {768}{5} a^{2} x^{5}+\frac {4192}{3} x^{12}-\frac {7424}{13} x^{13}+\frac {1280}{7} x^{14}-\frac {224}{5} x^{15}+\frac {1}{18} x^{18}-\frac {16}{17} x^{17}+8 x^{16}-\frac {29696}{11} x^{11}\) | \(223\) |
risch | \(4096 x^{8}-\frac {14336}{3} x^{9}-\frac {16384}{7} x^{7}+\frac {2048}{3} x^{6}+\frac {1}{2} a^{4} x^{2}+24 a^{2} x^{8}-1120 a \,x^{8}+\frac {1120}{11} x^{11} a -24 x^{12} a +\frac {2048}{3} x^{9} a -\frac {1536}{5} x^{10} a +\frac {9216}{7} x^{7} a +\frac {16}{5} x^{5} a^{3}+\frac {48}{13} x^{13} a -\frac {2}{7} x^{14} a +\frac {3}{5} x^{10} a^{2}-\frac {16}{3} x^{9} a^{2}-\frac {2}{3} x^{6} a^{3}+96 a^{2} x^{4}+4096 x^{10}+\frac {32}{3} a^{3} x^{3}-1024 a \,x^{6}-8 a^{3} x^{4}+\frac {2048}{5} a \,x^{5}-\frac {480}{7} a^{2} x^{7}+128 a^{2} x^{6}-\frac {768}{5} a^{2} x^{5}+\frac {4192}{3} x^{12}-\frac {7424}{13} x^{13}+\frac {1280}{7} x^{14}-\frac {224}{5} x^{15}+\frac {1}{18} x^{18}-\frac {16}{17} x^{17}+8 x^{16}-\frac {29696}{11} x^{11}\) | \(223\) |
default | \(\frac {x^{18}}{18}-\frac {16 x^{17}}{17}+8 x^{16}-\frac {224 x^{15}}{5}+\frac {\left (-4 a +2560\right ) x^{14}}{14}+\frac {\left (48 a -7424\right ) x^{13}}{13}+\frac {\left (-288 a +16768\right ) x^{12}}{12}+\frac {\left (1120 a -29696\right ) x^{11}}{11}+\frac {\left (2 a^{2}-2560 a +24576+\left (-2 a +128\right )^{2}\right ) x^{10}}{10}+\frac {\left (-16 a^{2}+3584 a -10240+2 \left (8 a -128\right ) \left (-2 a +128\right )\right ) x^{9}}{9}+\frac {\left (64 a^{2}-2560 a +2 \left (-16 a +64\right ) \left (-2 a +128\right )+\left (8 a -128\right )^{2}\right ) x^{8}}{8}+\frac {\left (-160 a^{2}+32 a \left (-2 a +128\right )+2 \left (-16 a +64\right ) \left (8 a -128\right )\right ) x^{7}}{7}+\frac {\left (2 a^{2} \left (-2 a +128\right )+32 a \left (8 a -128\right )+\left (-16 a +64\right )^{2}\right ) x^{6}}{6}+\frac {\left (2 a^{2} \left (8 a -128\right )+32 a \left (-16 a +64\right )\right ) x^{5}}{5}+\frac {\left (2 a^{2} \left (-16 a +64\right )+256 a^{2}\right ) x^{4}}{4}+\frac {32 a^{3} x^{3}}{3}+\frac {a^{4} x^{2}}{2}\) | \(267\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 182, normalized size = 0.87 \begin {gather*} \frac {1}{18} \, x^{18} - \frac {16}{17} \, x^{17} + 8 \, x^{16} - \frac {2}{7} \, {\left (a - 640\right )} x^{14} - \frac {224}{5} \, x^{15} + \frac {16}{13} \, {\left (3 \, a - 464\right )} x^{13} - \frac {8}{3} \, {\left (9 \, a - 524\right )} x^{12} + \frac {32}{11} \, {\left (35 \, a - 928\right )} x^{11} + \frac {1}{5} \, {\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{10} - \frac {16}{3} \, {\left (a^{2} - 128 \, a + 896\right )} x^{9} + 8 \, {\left (3 \, a^{2} - 140 \, a + 512\right )} x^{8} - \frac {32}{7} \, {\left (15 \, a^{2} - 288 \, a + 512\right )} x^{7} - \frac {2}{3} \, {\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{6} + \frac {1}{2} \, a^{4} x^{2} + \frac {32}{3} \, a^{3} x^{3} + \frac {16}{5} \, {\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{5} - 8 \, {\left (a^{3} - 12 \, a^{2}\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 182, normalized size = 0.87 \begin {gather*} \frac {1}{18} \, x^{18} - \frac {16}{17} \, x^{17} + 8 \, x^{16} - \frac {2}{7} \, {\left (a - 640\right )} x^{14} - \frac {224}{5} \, x^{15} + \frac {16}{13} \, {\left (3 \, a - 464\right )} x^{13} - \frac {8}{3} \, {\left (9 \, a - 524\right )} x^{12} + \frac {32}{11} \, {\left (35 \, a - 928\right )} x^{11} + \frac {1}{5} \, {\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{10} - \frac {16}{3} \, {\left (a^{2} - 128 \, a + 896\right )} x^{9} + 8 \, {\left (3 \, a^{2} - 140 \, a + 512\right )} x^{8} - \frac {32}{7} \, {\left (15 \, a^{2} - 288 \, a + 512\right )} x^{7} - \frac {2}{3} \, {\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{6} + \frac {1}{2} \, a^{4} x^{2} + \frac {32}{3} \, a^{3} x^{3} + \frac {16}{5} \, {\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{5} - 8 \, {\left (a^{3} - 12 \, a^{2}\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 212, normalized size = 1.01 \begin {gather*} \frac {a^{4} x^{2}}{2} + \frac {32 a^{3} x^{3}}{3} + \frac {x^{18}}{18} - \frac {16 x^{17}}{17} + 8 x^{16} - \frac {224 x^{15}}{5} + x^{14} \cdot \left (\frac {1280}{7} - \frac {2 a}{7}\right ) + x^{13} \cdot \left (\frac {48 a}{13} - \frac {7424}{13}\right ) + x^{12} \cdot \left (\frac {4192}{3} - 24 a\right ) + x^{11} \cdot \left (\frac {1120 a}{11} - \frac {29696}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2}}{5} - \frac {1536 a}{5} + 4096\right ) + x^{9} \left (- \frac {16 a^{2}}{3} + \frac {2048 a}{3} - \frac {14336}{3}\right ) + x^{8} \cdot \left (24 a^{2} - 1120 a + 4096\right ) + x^{7} \left (- \frac {480 a^{2}}{7} + \frac {9216 a}{7} - \frac {16384}{7}\right ) + x^{6} \left (- \frac {2 a^{3}}{3} + 128 a^{2} - 1024 a + \frac {2048}{3}\right ) + x^{5} \cdot \left (\frac {16 a^{3}}{5} - \frac {768 a^{2}}{5} + \frac {2048 a}{5}\right ) + x^{4} \left (- 8 a^{3} + 96 a^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.41, size = 222, normalized size = 1.06 \begin {gather*} \frac {1}{18} \, x^{18} - \frac {16}{17} \, x^{17} + 8 \, x^{16} - \frac {2}{7} \, a x^{14} - \frac {224}{5} \, x^{15} + \frac {48}{13} \, a x^{13} + \frac {1280}{7} \, x^{14} - 24 \, a x^{12} - \frac {7424}{13} \, x^{13} + \frac {3}{5} \, a^{2} x^{10} + \frac {1120}{11} \, a x^{11} + \frac {4192}{3} \, x^{12} - \frac {16}{3} \, a^{2} x^{9} - \frac {1536}{5} \, a x^{10} - \frac {29696}{11} \, x^{11} + 24 \, a^{2} x^{8} + \frac {2048}{3} \, a x^{9} + 4096 \, x^{10} - \frac {2}{3} \, a^{3} x^{6} - \frac {480}{7} \, a^{2} x^{7} - 1120 \, a x^{8} - \frac {14336}{3} \, x^{9} + \frac {16}{5} \, a^{3} x^{5} + 128 \, a^{2} x^{6} + \frac {9216}{7} \, a x^{7} + 4096 \, x^{8} - 8 \, a^{3} x^{4} - \frac {768}{5} \, a^{2} x^{5} - 1024 \, a x^{6} - \frac {16384}{7} \, x^{7} + \frac {1}{2} \, a^{4} x^{2} + \frac {32}{3} \, a^{3} x^{3} + 96 \, a^{2} x^{4} + \frac {2048}{5} \, a x^{5} + \frac {2048}{3} \, x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 178, normalized size = 0.85 \begin {gather*} x^{13}\,\left (\frac {48\,a}{13}-\frac {7424}{13}\right )-x^{12}\,\left (24\,a-\frac {4192}{3}\right )-x^{14}\,\left (\frac {2\,a}{7}-\frac {1280}{7}\right )+x^{11}\,\left (\frac {1120\,a}{11}-\frac {29696}{11}\right )+x^8\,\left (24\,a^2-1120\,a+4096\right )+x^{10}\,\left (\frac {3\,a^2}{5}-\frac {1536\,a}{5}+4096\right )-x^9\,\left (\frac {16\,a^2}{3}-\frac {2048\,a}{3}+\frac {14336}{3}\right )-x^7\,\left (\frac {480\,a^2}{7}-\frac {9216\,a}{7}+\frac {16384}{7}\right )-x^6\,\left (\frac {2\,a^3}{3}-128\,a^2+1024\,a-\frac {2048}{3}\right )-\frac {224\,x^{15}}{5}+8\,x^{16}-\frac {16\,x^{17}}{17}+\frac {x^{18}}{18}+\frac {32\,a^3\,x^3}{3}+\frac {a^4\,x^2}{2}+\frac {16\,a\,x^5\,\left (a^2-48\,a+128\right )}{5}-8\,a^2\,x^4\,\left (a-12\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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