Optimal. Leaf size=134 \[ \frac {a^3 x^2}{2}+8 a^2 x^3+6 (8-a) a x^4+\frac {4}{5} \left (128-96 a+3 a^2\right ) x^5-\frac {1}{2} \left (512-128 a+a^2\right ) x^6+\frac {48}{7} (48-5 a) x^7-4 (70-3 a) x^8+\frac {8}{3} (64-a) x^9-\frac {3}{10} (256-a) x^{10}+\frac {280 x^{11}}{11}-6 x^{12}+\frac {12 x^{13}}{13}-\frac {x^{14}}{14} \]
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Rubi [A]
time = 0.10, antiderivative size = 134, 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^3 x^2}{2}-\frac {1}{2} \left (a^2-128 a+512\right ) x^6+\frac {4}{5} \left (3 a^2-96 a+128\right ) x^5+8 a^2 x^3-\frac {3}{10} (256-a) x^{10}+\frac {8}{3} (64-a) x^9-4 (70-3 a) x^8+\frac {48}{7} (48-5 a) x^7+6 (8-a) a x^4-\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11} \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 )^3 \, dx &=\int \left (a^3 x+24 a^2 x^2-24 (-8+a) a x^3+4 \left (128-96 a+3 a^2\right ) x^4-3 \left (512-128 a+a^2\right ) x^5-48 (-48+5 a) x^6+32 (-70+3 a) x^7-24 (-64+a) x^8+3 (-256+a) x^9+280 x^{10}-72 x^{11}+12 x^{12}-x^{13}\right ) \, dx\\ &=\frac {a^3 x^2}{2}+8 a^2 x^3+6 (8-a) a x^4+\frac {4}{5} \left (128-96 a+3 a^2\right ) x^5-\frac {1}{2} \left (512-128 a+a^2\right ) x^6+\frac {48}{7} (48-5 a) x^7-4 (70-3 a) x^8+\frac {8}{3} (64-a) x^9-\frac {3}{10} (256-a) x^{10}+\frac {280 x^{11}}{11}-6 x^{12}+\frac {12 x^{13}}{13}-\frac {x^{14}}{14}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 130, normalized size = 0.97 \begin {gather*} \frac {a^3 x^2}{2}+8 a^2 x^3-6 (-8+a) a x^4+\frac {4}{5} \left (128-96 a+3 a^2\right ) x^5+\frac {1}{2} \left (-512+128 a-a^2\right ) x^6-\frac {48}{7} (-48+5 a) x^7+4 (-70+3 a) x^8-\frac {8}{3} (-64+a) x^9+\frac {3}{10} (-256+a) x^{10}+\frac {280 x^{11}}{11}-6 x^{12}+\frac {12 x^{13}}{13}-\frac {x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 143, normalized size = 1.07
method | result | size |
norman | \(\frac {a^{3} x^{2}}{2}+8 a^{2} x^{3}+\left (-6 a^{2}+48 a \right ) x^{4}+\left (\frac {12}{5} a^{2}-\frac {384}{5} a +\frac {512}{5}\right ) x^{5}+\left (-\frac {1}{2} a^{2}+64 a -256\right ) x^{6}+\left (-\frac {240 a}{7}+\frac {2304}{7}\right ) x^{7}+\left (12 a -280\right ) x^{8}+\left (-\frac {8 a}{3}+\frac {512}{3}\right ) x^{9}+\left (\frac {3 a}{10}-\frac {384}{5}\right ) x^{10}+\frac {280 x^{11}}{11}-6 x^{12}+\frac {12 x^{13}}{13}-\frac {x^{14}}{14}\) | \(115\) |
gosper | \(\frac {1}{2} a^{3} x^{2}+8 a^{2} x^{3}-6 a^{2} x^{4}+48 a \,x^{4}+\frac {12}{5} a^{2} x^{5}-\frac {384}{5} a \,x^{5}+\frac {512}{5} x^{5}-\frac {1}{2} a^{2} x^{6}+64 a \,x^{6}-256 x^{6}-\frac {240}{7} x^{7} a +\frac {2304}{7} x^{7}+12 a \,x^{8}-280 x^{8}-\frac {8}{3} x^{9} a +\frac {512}{3} x^{9}+\frac {3}{10} x^{10} a -\frac {384}{5} x^{10}+\frac {280}{11} x^{11}-6 x^{12}+\frac {12}{13} x^{13}-\frac {1}{14} x^{14}\) | \(134\) |
risch | \(\frac {1}{2} a^{3} x^{2}+8 a^{2} x^{3}-6 a^{2} x^{4}+48 a \,x^{4}+\frac {12}{5} a^{2} x^{5}-\frac {384}{5} a \,x^{5}+\frac {512}{5} x^{5}-\frac {1}{2} a^{2} x^{6}+64 a \,x^{6}-256 x^{6}-\frac {240}{7} x^{7} a +\frac {2304}{7} x^{7}+12 a \,x^{8}-280 x^{8}-\frac {8}{3} x^{9} a +\frac {512}{3} x^{9}+\frac {3}{10} x^{10} a -\frac {384}{5} x^{10}+\frac {280}{11} x^{11}-6 x^{12}+\frac {12}{13} x^{13}-\frac {1}{14} x^{14}\) | \(134\) |
default | \(-\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11}+\frac {\left (3 a -768\right ) x^{10}}{10}+\frac {\left (-24 a +1536\right ) x^{9}}{9}+\frac {\left (96 a -2240\right ) x^{8}}{8}+\frac {\left (-240 a +2304\right ) x^{7}}{7}+\frac {\left (a \left (-2 a +128\right )+256 a -1536-a^{2}\right ) x^{6}}{6}+\frac {\left (a \left (8 a -128\right )-256 a +512+4 a^{2}\right ) x^{5}}{5}+\frac {\left (a \left (-16 a +64\right )+128 a -8 a^{2}\right ) x^{4}}{4}+8 a^{2} x^{3}+\frac {a^{3} x^{2}}{2}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 113, normalized size = 0.84 \begin {gather*} -\frac {1}{14} \, x^{14} + \frac {12}{13} \, x^{13} - 6 \, x^{12} + \frac {3}{10} \, {\left (a - 256\right )} x^{10} + \frac {280}{11} \, x^{11} - \frac {8}{3} \, {\left (a - 64\right )} x^{9} + 4 \, {\left (3 \, a - 70\right )} x^{8} - \frac {48}{7} \, {\left (5 \, a - 48\right )} x^{7} - \frac {1}{2} \, {\left (a^{2} - 128 \, a + 512\right )} x^{6} + \frac {4}{5} \, {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{5} + \frac {1}{2} \, a^{3} x^{2} + 8 \, a^{2} x^{3} - 6 \, {\left (a^{2} - 8 \, a\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 113, normalized size = 0.84 \begin {gather*} -\frac {1}{14} \, x^{14} + \frac {12}{13} \, x^{13} - 6 \, x^{12} + \frac {3}{10} \, {\left (a - 256\right )} x^{10} + \frac {280}{11} \, x^{11} - \frac {8}{3} \, {\left (a - 64\right )} x^{9} + 4 \, {\left (3 \, a - 70\right )} x^{8} - \frac {48}{7} \, {\left (5 \, a - 48\right )} x^{7} - \frac {1}{2} \, {\left (a^{2} - 128 \, a + 512\right )} x^{6} + \frac {4}{5} \, {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{5} + \frac {1}{2} \, a^{3} x^{2} + 8 \, a^{2} x^{3} - 6 \, {\left (a^{2} - 8 \, a\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 128, normalized size = 0.96 \begin {gather*} \frac {a^{3} x^{2}}{2} + 8 a^{2} x^{3} - \frac {x^{14}}{14} + \frac {12 x^{13}}{13} - 6 x^{12} + \frac {280 x^{11}}{11} + x^{10} \cdot \left (\frac {3 a}{10} - \frac {384}{5}\right ) + x^{9} \cdot \left (\frac {512}{3} - \frac {8 a}{3}\right ) + x^{8} \cdot \left (12 a - 280\right ) + x^{7} \cdot \left (\frac {2304}{7} - \frac {240 a}{7}\right ) + x^{6} \left (- \frac {a^{2}}{2} + 64 a - 256\right ) + x^{5} \cdot \left (\frac {12 a^{2}}{5} - \frac {384 a}{5} + \frac {512}{5}\right ) + x^{4} \left (- 6 a^{2} + 48 a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.38, size = 133, normalized size = 0.99 \begin {gather*} -\frac {1}{14} \, x^{14} + \frac {12}{13} \, x^{13} - 6 \, x^{12} + \frac {3}{10} \, a x^{10} + \frac {280}{11} \, x^{11} - \frac {8}{3} \, a x^{9} - \frac {384}{5} \, x^{10} + 12 \, a x^{8} + \frac {512}{3} \, x^{9} - \frac {1}{2} \, a^{2} x^{6} - \frac {240}{7} \, a x^{7} - 280 \, x^{8} + \frac {12}{5} \, a^{2} x^{5} + 64 \, a x^{6} + \frac {2304}{7} \, x^{7} - 6 \, a^{2} x^{4} - \frac {384}{5} \, a x^{5} - 256 \, x^{6} + \frac {1}{2} \, a^{3} x^{2} + 8 \, a^{2} x^{3} + 48 \, a x^{4} + \frac {512}{5} \, x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.12, size = 113, normalized size = 0.84 \begin {gather*} x^8\,\left (12\,a-280\right )+x^{10}\,\left (\frac {3\,a}{10}-\frac {384}{5}\right )-x^9\,\left (\frac {8\,a}{3}-\frac {512}{3}\right )-x^7\,\left (\frac {240\,a}{7}-\frac {2304}{7}\right )-x^6\,\left (\frac {a^2}{2}-64\,a+256\right )+x^5\,\left (\frac {12\,a^2}{5}-\frac {384\,a}{5}+\frac {512}{5}\right )+\frac {280\,x^{11}}{11}-6\,x^{12}+\frac {12\,x^{13}}{13}-\frac {x^{14}}{14}+8\,a^2\,x^3+\frac {a^3\,x^2}{2}-6\,a\,x^4\,\left (a-8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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