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 = 0.05, 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, 270} \begin {gather*} \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 {a^6 x^9}{9}+\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 270
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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.71, size = 68, normalized size = 0.83 \begin {gather*} \frac {1}{21} x^{21} b^{6} + \frac {6}{19} x^{19} b^{5} a + \frac {15}{17} x^{17} b^{4} a^{2} + \frac {4}{3} x^{15} b^{3} a^{3} + \frac {15}{13} x^{13} b^{2} a^{4} + \frac {6}{11} x^{11} b a^{5} + \frac {1}{9} x^{9} a^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, 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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maple [A] time = 0.00, size = 69, normalized size = 0.84 \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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maxima [A] time = 1.35, 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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sympy [A] time = 0.09, 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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