Integrand size = 20, antiderivative size = 95 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {a^2 (A b-a B) \left (a+b x^2\right )^6}{12 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^7}{14 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^8}{16 b^4}+\frac {B \left (a+b x^2\right )^9}{18 b^4} \]
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Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {a^2 \left (a+b x^2\right )^6 (A b-a B)}{12 b^4}+\frac {\left (a+b x^2\right )^8 (A b-3 a B)}{16 b^4}-\frac {a \left (a+b x^2\right )^7 (2 A b-3 a B)}{14 b^4}+\frac {B \left (a+b x^2\right )^9}{18 b^4} \]
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Rule 77
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^5 (A+B x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac {(A b-3 a B) (a+b x)^7}{b^3}+\frac {B (a+b x)^8}{b^3}\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 (A b-a B) \left (a+b x^2\right )^6}{12 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^7}{14 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^8}{16 b^4}+\frac {B \left (a+b x^2\right )^9}{18 b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {x^6 \left (168 a^5 A+126 a^4 (5 A b+a B) x^2+504 a^3 b (2 A b+a B) x^4+840 a^2 b^2 (A b+a B) x^6+360 a b^3 (A b+2 a B) x^8+63 b^4 (A b+5 a B) x^{10}+56 b^5 B x^{12}\right )}{1008} \]
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Time = 2.50 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.26
method | result | size |
norman | \(\frac {a^{5} A \,x^{6}}{6}+\left (\frac {5}{8} a^{4} b A +\frac {1}{8} a^{5} B \right ) x^{8}+\left (a^{3} b^{2} A +\frac {1}{2} a^{4} b B \right ) x^{10}+\left (\frac {5}{6} a^{2} b^{3} A +\frac {5}{6} a^{3} b^{2} B \right ) x^{12}+\left (\frac {5}{14} a \,b^{4} A +\frac {5}{7} a^{2} b^{3} B \right ) x^{14}+\left (\frac {1}{16} b^{5} A +\frac {5}{16} a \,b^{4} B \right ) x^{16}+\frac {b^{5} B \,x^{18}}{18}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{18}}{18}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{16}}{16}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{14}}{14}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{12}}{12}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{10}}{10}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{8}}{8}+\frac {a^{5} A \,x^{6}}{6}\) | \(124\) |
gosper | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
risch | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
parallelrisch | \(\frac {1}{6} a^{5} A \,x^{6}+\frac {5}{8} x^{8} a^{4} b A +\frac {1}{8} x^{8} a^{5} B +x^{10} a^{3} b^{2} A +\frac {1}{2} x^{10} a^{4} b B +\frac {5}{6} x^{12} a^{2} b^{3} A +\frac {5}{6} x^{12} a^{3} b^{2} B +\frac {5}{14} x^{14} a \,b^{4} A +\frac {5}{7} x^{14} a^{2} b^{3} B +\frac {1}{16} x^{16} b^{5} A +\frac {5}{16} x^{16} a \,b^{4} B +\frac {1}{18} b^{5} B \,x^{18}\) | \(125\) |
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Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{16} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac {5}{14} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac {5}{6} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \]
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Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{18}}{18} + x^{16} \left (\frac {A b^{5}}{16} + \frac {5 B a b^{4}}{16}\right ) + x^{14} \cdot \left (\frac {5 A a b^{4}}{14} + \frac {5 B a^{2} b^{3}}{7}\right ) + x^{12} \cdot \left (\frac {5 A a^{2} b^{3}}{6} + \frac {5 B a^{3} b^{2}}{6}\right ) + x^{10} \left (A a^{3} b^{2} + \frac {B a^{4} b}{2}\right ) + x^{8} \cdot \left (\frac {5 A a^{4} b}{8} + \frac {B a^{5}}{8}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {1}{16} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac {5}{14} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac {5}{6} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac {1}{8} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \]
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Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{18} \, B b^{5} x^{18} + \frac {5}{16} \, B a b^{4} x^{16} + \frac {1}{16} \, A b^{5} x^{16} + \frac {5}{7} \, B a^{2} b^{3} x^{14} + \frac {5}{14} \, A a b^{4} x^{14} + \frac {5}{6} \, B a^{3} b^{2} x^{12} + \frac {5}{6} \, A a^{2} b^{3} x^{12} + \frac {1}{2} \, B a^{4} b x^{10} + A a^{3} b^{2} x^{10} + \frac {1}{8} \, B a^{5} x^{8} + \frac {5}{8} \, A a^{4} b x^{8} + \frac {1}{6} \, A a^{5} x^{6} \]
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Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^8\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^{16}\,\left (\frac {A\,b^5}{16}+\frac {5\,B\,a\,b^4}{16}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{18}}{18}+\frac {5\,a^2\,b^2\,x^{12}\,\left (A\,b+B\,a\right )}{6}+\frac {a^3\,b\,x^{10}\,\left (2\,A\,b+B\,a\right )}{2}+\frac {5\,a\,b^3\,x^{14}\,\left (A\,b+2\,B\,a\right )}{14} \]
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