Optimal. Leaf size=117 \[ \frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{13} a^3 b x^{13} (a B+2 A b)+\frac {2}{3} a^2 b^2 x^{15} (a B+A b)+\frac {1}{19} b^4 x^{19} (5 a B+A b)+\frac {5}{17} a b^3 x^{17} (2 a B+A b)+\frac {1}{21} b^5 B x^{21} \]
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Rubi [A] time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ \frac {2}{3} a^2 b^2 x^{15} (a B+A b)+\frac {5}{13} a^3 b x^{13} (a B+2 A b)+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {1}{9} a^5 A x^9+\frac {1}{19} b^4 x^{19} (5 a B+A b)+\frac {5}{17} a b^3 x^{17} (2 a B+A b)+\frac {1}{21} b^5 B x^{21} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\int \left (a^5 A x^8+a^4 (5 A b+a B) x^{10}+5 a^3 b (2 A b+a B) x^{12}+10 a^2 b^2 (A b+a B) x^{14}+5 a b^3 (A b+2 a B) x^{16}+b^4 (A b+5 a B) x^{18}+b^5 B x^{20}\right ) \, dx\\ &=\frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {2}{3} a^2 b^2 (A b+a B) x^{15}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{21} b^5 B x^{21}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 117, normalized size = 1.00 \[ \frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{13} a^3 b x^{13} (a B+2 A b)+\frac {2}{3} a^2 b^2 x^{15} (a B+A b)+\frac {1}{19} b^4 x^{19} (5 a B+A b)+\frac {5}{17} a b^3 x^{17} (2 a B+A b)+\frac {1}{21} b^5 B x^{21} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 125, normalized size = 1.07 \[ \frac {1}{21} x^{21} b^{5} B + \frac {5}{19} x^{19} b^{4} a B + \frac {1}{19} x^{19} b^{5} A + \frac {10}{17} x^{17} b^{3} a^{2} B + \frac {5}{17} x^{17} b^{4} a A + \frac {2}{3} x^{15} b^{2} a^{3} B + \frac {2}{3} x^{15} b^{3} a^{2} A + \frac {5}{13} x^{13} b a^{4} B + \frac {10}{13} x^{13} b^{2} a^{3} A + \frac {1}{11} x^{11} a^{5} B + \frac {5}{11} x^{11} b a^{4} A + \frac {1}{9} x^{9} a^{5} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 125, normalized size = 1.07 \[ \frac {1}{21} \, B b^{5} x^{21} + \frac {5}{19} \, B a b^{4} x^{19} + \frac {1}{19} \, A b^{5} x^{19} + \frac {10}{17} \, B a^{2} b^{3} x^{17} + \frac {5}{17} \, A a b^{4} x^{17} + \frac {2}{3} \, B a^{3} b^{2} x^{15} + \frac {2}{3} \, A a^{2} b^{3} x^{15} + \frac {5}{13} \, B a^{4} b x^{13} + \frac {10}{13} \, A a^{3} b^{2} x^{13} + \frac {1}{11} \, B a^{5} x^{11} + \frac {5}{11} \, A a^{4} b x^{11} + \frac {1}{9} \, A a^{5} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 124, normalized size = 1.06 \[ \frac {B \,b^{5} x^{21}}{21}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{19}}{19}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{17}}{17}+\frac {A \,a^{5} x^{9}}{9}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{15}}{15}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{13}}{13}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{11}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 119, normalized size = 1.02 \[ \frac {1}{21} \, B b^{5} x^{21} + \frac {1}{19} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac {5}{17} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac {2}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac {1}{9} \, A a^{5} x^{9} + \frac {5}{13} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 107, normalized size = 0.91 \[ x^{11}\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^{19}\,\left (\frac {A\,b^5}{19}+\frac {5\,B\,a\,b^4}{19}\right )+\frac {A\,a^5\,x^9}{9}+\frac {B\,b^5\,x^{21}}{21}+\frac {2\,a^2\,b^2\,x^{15}\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^{13}\,\left (2\,A\,b+B\,a\right )}{13}+\frac {5\,a\,b^3\,x^{17}\,\left (A\,b+2\,B\,a\right )}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 138, normalized size = 1.18 \[ \frac {A a^{5} x^{9}}{9} + \frac {B b^{5} x^{21}}{21} + x^{19} \left (\frac {A b^{5}}{19} + \frac {5 B a b^{4}}{19}\right ) + x^{17} \left (\frac {5 A a b^{4}}{17} + \frac {10 B a^{2} b^{3}}{17}\right ) + x^{15} \left (\frac {2 A a^{2} b^{3}}{3} + \frac {2 B a^{3} b^{2}}{3}\right ) + x^{13} \left (\frac {10 A a^{3} b^{2}}{13} + \frac {5 B a^{4} b}{13}\right ) + x^{11} \left (\frac {5 A a^{4} b}{11} + \frac {B a^{5}}{11}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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