Optimal. Leaf size=82 \[ \frac {a^6 x^9}{9}+\frac {6}{11} a^5 b x^{11}+\frac {15}{13} a^4 b^2 x^{13}+\frac {4}{3} a^3 b^3 x^{15}+\frac {15}{17} a^2 b^4 x^{17}+\frac {6}{19} a b^5 x^{19}+\frac {b^6 x^{21}}{21} \]
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Rubi [A]
time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276}
\begin {gather*} \frac {a^6 x^9}{9}+\frac {6}{11} a^5 b x^{11}+\frac {15}{13} a^4 b^2 x^{13}+\frac {4}{3} a^3 b^3 x^{15}+\frac {15}{17} a^2 b^4 x^{17}+\frac {6}{19} a b^5 x^{19}+\frac {b^6 x^{21}}{21} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 276
Rubi steps
\begin {align*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^8 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 x^8+6 a^5 b^7 x^{10}+15 a^4 b^8 x^{12}+20 a^3 b^9 x^{14}+15 a^2 b^{10} x^{16}+6 a b^{11} x^{18}+b^{12} x^{20}\right ) \, dx}{b^6}\\ &=\frac {a^6 x^9}{9}+\frac {6}{11} a^5 b x^{11}+\frac {15}{13} a^4 b^2 x^{13}+\frac {4}{3} a^3 b^3 x^{15}+\frac {15}{17} a^2 b^4 x^{17}+\frac {6}{19} a b^5 x^{19}+\frac {b^6 x^{21}}{21}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 82, normalized size = 1.00 \begin {gather*} \frac {a^6 x^9}{9}+\frac {6}{11} a^5 b x^{11}+\frac {15}{13} a^4 b^2 x^{13}+\frac {4}{3} a^3 b^3 x^{15}+\frac {15}{17} a^2 b^4 x^{17}+\frac {6}{19} a b^5 x^{19}+\frac {b^6 x^{21}}{21} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 69, normalized size = 0.84
method | result | size |
default | \(\frac {1}{9} a^{6} x^{9}+\frac {6}{11} a^{5} b \,x^{11}+\frac {15}{13} a^{4} b^{2} x^{13}+\frac {4}{3} a^{3} b^{3} x^{15}+\frac {15}{17} a^{2} b^{4} x^{17}+\frac {6}{19} a \,b^{5} x^{19}+\frac {1}{21} b^{6} x^{21}\) | \(69\) |
norman | \(\frac {1}{9} a^{6} x^{9}+\frac {6}{11} a^{5} b \,x^{11}+\frac {15}{13} a^{4} b^{2} x^{13}+\frac {4}{3} a^{3} b^{3} x^{15}+\frac {15}{17} a^{2} b^{4} x^{17}+\frac {6}{19} a \,b^{5} x^{19}+\frac {1}{21} b^{6} x^{21}\) | \(69\) |
risch | \(\frac {1}{9} a^{6} x^{9}+\frac {6}{11} a^{5} b \,x^{11}+\frac {15}{13} a^{4} b^{2} x^{13}+\frac {4}{3} a^{3} b^{3} x^{15}+\frac {15}{17} a^{2} b^{4} x^{17}+\frac {6}{19} a \,b^{5} x^{19}+\frac {1}{21} b^{6} x^{21}\) | \(69\) |
gosper | \(\frac {x^{9} \left (138567 b^{6} x^{12}+918918 a \,b^{5} x^{10}+2567565 a^{2} b^{4} x^{8}+3879876 a^{3} b^{3} x^{6}+3357585 a^{4} b^{2} x^{4}+1587222 a^{5} b \,x^{2}+323323 a^{6}\right )}{2909907}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{21} \, b^{6} x^{21} + \frac {6}{19} \, a b^{5} x^{19} + \frac {15}{17} \, a^{2} b^{4} x^{17} + \frac {4}{3} \, a^{3} b^{3} x^{15} + \frac {15}{13} \, a^{4} b^{2} x^{13} + \frac {6}{11} \, a^{5} b x^{11} + \frac {1}{9} \, a^{6} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{21} \, b^{6} x^{21} + \frac {6}{19} \, a b^{5} x^{19} + \frac {15}{17} \, a^{2} b^{4} x^{17} + \frac {4}{3} \, a^{3} b^{3} x^{15} + \frac {15}{13} \, a^{4} b^{2} x^{13} + \frac {6}{11} \, a^{5} b x^{11} + \frac {1}{9} \, a^{6} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 80, normalized size = 0.98 \begin {gather*} \frac {a^{6} x^{9}}{9} + \frac {6 a^{5} b x^{11}}{11} + \frac {15 a^{4} b^{2} x^{13}}{13} + \frac {4 a^{3} b^{3} x^{15}}{3} + \frac {15 a^{2} b^{4} x^{17}}{17} + \frac {6 a b^{5} x^{19}}{19} + \frac {b^{6} x^{21}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.32, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{21} \, b^{6} x^{21} + \frac {6}{19} \, a b^{5} x^{19} + \frac {15}{17} \, a^{2} b^{4} x^{17} + \frac {4}{3} \, a^{3} b^{3} x^{15} + \frac {15}{13} \, a^{4} b^{2} x^{13} + \frac {6}{11} \, a^{5} b x^{11} + \frac {1}{9} \, a^{6} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 68, normalized size = 0.83 \begin {gather*} \frac {a^6\,x^9}{9}+\frac {6\,a^5\,b\,x^{11}}{11}+\frac {15\,a^4\,b^2\,x^{13}}{13}+\frac {4\,a^3\,b^3\,x^{15}}{3}+\frac {15\,a^2\,b^4\,x^{17}}{17}+\frac {6\,a\,b^5\,x^{19}}{19}+\frac {b^6\,x^{21}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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