Optimal. Leaf size=117 \[ \frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \]
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Rubi [A]
time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77}
\begin {gather*} \frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 x^{12} (a B+5 A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {5}{8} a^2 b^2 x^{16} (a B+A b)+\frac {1}{20} b^4 x^{20} (5 a B+A b)+\frac {5}{18} a b^3 x^{18} (2 a B+A b)+\frac {1}{22} b^5 B x^{22} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rubi steps
\begin {align*} \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x^4 (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^5 A x^4+a^4 (5 A b+a B) x^5+5 a^3 b (2 A b+a B) x^6+10 a^2 b^2 (A b+a B) x^7+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^9+b^5 B x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 124, normalized size = 1.06
method | result | size |
norman | \(\frac {a^{5} A \,x^{10}}{10}+\left (\frac {5}{12} a^{4} b A +\frac {1}{12} a^{5} B \right ) x^{12}+\left (\frac {5}{7} a^{3} b^{2} A +\frac {5}{14} a^{4} b B \right ) x^{14}+\left (\frac {5}{8} a^{2} b^{3} A +\frac {5}{8} a^{3} b^{2} B \right ) x^{16}+\left (\frac {5}{18} a \,b^{4} A +\frac {5}{9} a^{2} b^{3} B \right ) x^{18}+\left (\frac {1}{20} b^{5} A +\frac {1}{4} a \,b^{4} B \right ) x^{20}+\frac {b^{5} B \,x^{22}}{22}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{22}}{22}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{20}}{20}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{18}}{18}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{16}}{16}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{14}}{14}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{12}}{12}+\frac {a^{5} A \,x^{10}}{10}\) | \(124\) |
gosper | \(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(126\) |
risch | \(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 119, normalized size = 1.02 \begin {gather*} \frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.88, size = 119, normalized size = 1.02 \begin {gather*} \frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 136, normalized size = 1.16 \begin {gather*} \frac {A a^{5} x^{10}}{10} + \frac {B b^{5} x^{22}}{22} + x^{20} \left (\frac {A b^{5}}{20} + \frac {B a b^{4}}{4}\right ) + x^{18} \cdot \left (\frac {5 A a b^{4}}{18} + \frac {5 B a^{2} b^{3}}{9}\right ) + x^{16} \cdot \left (\frac {5 A a^{2} b^{3}}{8} + \frac {5 B a^{3} b^{2}}{8}\right ) + x^{14} \cdot \left (\frac {5 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{14}\right ) + x^{12} \cdot \left (\frac {5 A a^{4} b}{12} + \frac {B a^{5}}{12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 125, normalized size = 1.07 \begin {gather*} \frac {1}{22} \, B b^{5} x^{22} + \frac {1}{4} \, B a b^{4} x^{20} + \frac {1}{20} \, A b^{5} x^{20} + \frac {5}{9} \, B a^{2} b^{3} x^{18} + \frac {5}{18} \, A a b^{4} x^{18} + \frac {5}{8} \, B a^{3} b^{2} x^{16} + \frac {5}{8} \, A a^{2} b^{3} x^{16} + \frac {5}{14} \, B a^{4} b x^{14} + \frac {5}{7} \, A a^{3} b^{2} x^{14} + \frac {1}{12} \, B a^{5} x^{12} + \frac {5}{12} \, A a^{4} b x^{12} + \frac {1}{10} \, A a^{5} x^{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 107, normalized size = 0.91 \begin {gather*} x^{12}\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{20}\,\left (\frac {A\,b^5}{20}+\frac {B\,a\,b^4}{4}\right )+\frac {A\,a^5\,x^{10}}{10}+\frac {B\,b^5\,x^{22}}{22}+\frac {5\,a^2\,b^2\,x^{16}\,\left (A\,b+B\,a\right )}{8}+\frac {5\,a^3\,b\,x^{14}\,\left (2\,A\,b+B\,a\right )}{14}+\frac {5\,a\,b^3\,x^{18}\,\left (A\,b+2\,B\,a\right )}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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