Optimal. Leaf size=73 \[ \frac {1}{19} (x+1)^{19}-\frac {7}{18} (x+1)^{18}+\frac {21}{17} (x+1)^{17}-\frac {35}{16} (x+1)^{16}+\frac {7}{3} (x+1)^{15}-\frac {3}{2} (x+1)^{14}+\frac {7}{13} (x+1)^{13}-\frac {1}{12} (x+1)^{12} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 43} \begin {gather*} \frac {1}{19} (x+1)^{19}-\frac {7}{18} (x+1)^{18}+\frac {21}{17} (x+1)^{17}-\frac {35}{16} (x+1)^{16}+\frac {7}{3} (x+1)^{15}-\frac {3}{2} (x+1)^{14}+\frac {7}{13} (x+1)^{13}-\frac {1}{12} (x+1)^{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+7 (1+x)^{12}-21 (1+x)^{13}+35 (1+x)^{14}-35 (1+x)^{15}+21 (1+x)^{16}-7 (1+x)^{17}+(1+x)^{18}\right ) \, dx\\ &=-\frac {1}{12} (1+x)^{12}+\frac {7}{13} (1+x)^{13}-\frac {3}{2} (1+x)^{14}+\frac {7}{3} (1+x)^{15}-\frac {35}{16} (1+x)^{16}+\frac {21}{17} (1+x)^{17}-\frac {7}{18} (1+x)^{18}+\frac {1}{19} (1+x)^{19}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 79, normalized size = 1.08 \begin {gather*} \frac {x^{19}}{19}+\frac {11 x^{18}}{18}+\frac {55 x^{17}}{17}+\frac {165 x^{16}}{16}+22 x^{15}+33 x^{14}+\frac {462 x^{13}}{13}+\frac {55 x^{12}}{2}+15 x^{11}+\frac {11 x^{10}}{2}+\frac {11 x^9}{9}+\frac {x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} x^{19} + \frac {11}{18} x^{18} + \frac {55}{17} x^{17} + \frac {165}{16} x^{16} + 22 x^{15} + 33 x^{14} + \frac {462}{13} x^{13} + \frac {55}{2} x^{12} + 15 x^{11} + \frac {11}{2} x^{10} + \frac {11}{9} x^{9} + \frac {1}{8} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 0.85 \begin {gather*} \frac {1}{19} x^{19}+\frac {11}{18} x^{18}+\frac {55}{17} x^{17}+\frac {165}{16} x^{16}+22 x^{15}+33 x^{14}+\frac {462}{13} x^{13}+\frac {55}{2} x^{12}+15 x^{11}+\frac {11}{2} x^{10}+\frac {11}{9} x^{9}+\frac {1}{8} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 61, normalized size = 0.84 \begin {gather*} \frac {x^{19}}{19}+\frac {11\,x^{18}}{18}+\frac {55\,x^{17}}{17}+\frac {165\,x^{16}}{16}+22\,x^{15}+33\,x^{14}+\frac {462\,x^{13}}{13}+\frac {55\,x^{12}}{2}+15\,x^{11}+\frac {11\,x^{10}}{2}+\frac {11\,x^9}{9}+\frac {x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 71, normalized size = 0.97 \begin {gather*} \frac {x^{19}}{19} + \frac {11 x^{18}}{18} + \frac {55 x^{17}}{17} + \frac {165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac {462 x^{13}}{13} + \frac {55 x^{12}}{2} + 15 x^{11} + \frac {11 x^{10}}{2} + \frac {11 x^{9}}{9} + \frac {x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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