Integrand size = 20, antiderivative size = 117 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{9} a^3 b (2 A b+a B) x^9+\frac {10}{11} a^2 b^2 (A b+a B) x^{11}+\frac {5}{13} a b^3 (A b+2 a B) x^{13}+\frac {1}{15} b^4 (A b+5 a B) x^{15}+\frac {1}{17} b^5 B x^{17} \]
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Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {5}{9} a^3 b x^9 (a B+2 A b)+\frac {10}{11} a^2 b^2 x^{11} (a B+A b)+\frac {1}{15} b^4 x^{15} (5 a B+A b)+\frac {5}{13} a b^3 x^{13} (2 a B+A b)+\frac {1}{17} b^5 B x^{17} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 A x^4+a^4 (5 A b+a B) x^6+5 a^3 b (2 A b+a B) x^8+10 a^2 b^2 (A b+a B) x^{10}+5 a b^3 (A b+2 a B) x^{12}+b^4 (A b+5 a B) x^{14}+b^5 B x^{16}\right ) \, dx \\ & = \frac {1}{5} a^5 A x^5+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{9} a^3 b (2 A b+a B) x^9+\frac {10}{11} a^2 b^2 (A b+a B) x^{11}+\frac {5}{13} a b^3 (A b+2 a B) x^{13}+\frac {1}{15} b^4 (A b+5 a B) x^{15}+\frac {1}{17} b^5 B x^{17} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{9} a^3 b (2 A b+a B) x^9+\frac {10}{11} a^2 b^2 (A b+a B) x^{11}+\frac {5}{13} a b^3 (A b+2 a B) x^{13}+\frac {1}{15} b^4 (A b+5 a B) x^{15}+\frac {1}{17} b^5 B x^{17} \]
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Time = 2.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {a^{5} A \,x^{5}}{5}+\left (\frac {5}{7} a^{4} b A +\frac {1}{7} a^{5} B \right ) x^{7}+\left (\frac {10}{9} a^{3} b^{2} A +\frac {5}{9} a^{4} b B \right ) x^{9}+\left (\frac {10}{11} a^{2} b^{3} A +\frac {10}{11} a^{3} b^{2} B \right ) x^{11}+\left (\frac {5}{13} a \,b^{4} A +\frac {10}{13} a^{2} b^{3} B \right ) x^{13}+\left (\frac {1}{15} b^{5} A +\frac {1}{3} a \,b^{4} B \right ) x^{15}+\frac {b^{5} B \,x^{17}}{17}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{17}}{17}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{15}}{15}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{13}}{13}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{11}}{11}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{9}}{9}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{7}}{7}+\frac {a^{5} A \,x^{5}}{5}\) | \(124\) |
gosper | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {10}{9} x^{9} a^{3} b^{2} A +\frac {5}{9} x^{9} a^{4} b B +\frac {10}{11} x^{11} a^{2} b^{3} A +\frac {10}{11} x^{11} a^{3} b^{2} B +\frac {5}{13} x^{13} a \,b^{4} A +\frac {10}{13} x^{13} a^{2} b^{3} B +\frac {1}{15} x^{15} b^{5} A +\frac {1}{3} x^{15} a \,b^{4} B +\frac {1}{17} b^{5} B \,x^{17}\) | \(126\) |
risch | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {10}{9} x^{9} a^{3} b^{2} A +\frac {5}{9} x^{9} a^{4} b B +\frac {10}{11} x^{11} a^{2} b^{3} A +\frac {10}{11} x^{11} a^{3} b^{2} B +\frac {5}{13} x^{13} a \,b^{4} A +\frac {10}{13} x^{13} a^{2} b^{3} B +\frac {1}{15} x^{15} b^{5} A +\frac {1}{3} x^{15} a \,b^{4} B +\frac {1}{17} b^{5} B \,x^{17}\) | \(126\) |
parallelrisch | \(\frac {1}{5} a^{5} A \,x^{5}+\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {10}{9} x^{9} a^{3} b^{2} A +\frac {5}{9} x^{9} a^{4} b B +\frac {10}{11} x^{11} a^{2} b^{3} A +\frac {10}{11} x^{11} a^{3} b^{2} B +\frac {5}{13} x^{13} a \,b^{4} A +\frac {10}{13} x^{13} a^{2} b^{3} B +\frac {1}{15} x^{15} b^{5} A +\frac {1}{3} x^{15} a \,b^{4} B +\frac {1}{17} b^{5} B \,x^{17}\) | \(126\) |
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Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{17} \, B b^{5} x^{17} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{13} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{13} + \frac {10}{11} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {5}{9} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
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Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{5}}{5} + \frac {B b^{5} x^{17}}{17} + x^{15} \left (\frac {A b^{5}}{15} + \frac {B a b^{4}}{3}\right ) + x^{13} \cdot \left (\frac {5 A a b^{4}}{13} + \frac {10 B a^{2} b^{3}}{13}\right ) + x^{11} \cdot \left (\frac {10 A a^{2} b^{3}}{11} + \frac {10 B a^{3} b^{2}}{11}\right ) + x^{9} \cdot \left (\frac {10 A a^{3} b^{2}}{9} + \frac {5 B a^{4} b}{9}\right ) + x^{7} \cdot \left (\frac {5 A a^{4} b}{7} + \frac {B a^{5}}{7}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{17} \, B b^{5} x^{17} + \frac {1}{15} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + \frac {5}{13} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{13} + \frac {10}{11} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {5}{9} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{9} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
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Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{17} \, B b^{5} x^{17} + \frac {1}{3} \, B a b^{4} x^{15} + \frac {1}{15} \, A b^{5} x^{15} + \frac {10}{13} \, B a^{2} b^{3} x^{13} + \frac {5}{13} \, A a b^{4} x^{13} + \frac {10}{11} \, B a^{3} b^{2} x^{11} + \frac {10}{11} \, A a^{2} b^{3} x^{11} + \frac {5}{9} \, B a^{4} b x^{9} + \frac {10}{9} \, A a^{3} b^{2} x^{9} + \frac {1}{7} \, B a^{5} x^{7} + \frac {5}{7} \, A a^{4} b x^{7} + \frac {1}{5} \, A a^{5} x^{5} \]
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Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^4 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^7\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^{15}\,\left (\frac {A\,b^5}{15}+\frac {B\,a\,b^4}{3}\right )+\frac {A\,a^5\,x^5}{5}+\frac {B\,b^5\,x^{17}}{17}+\frac {10\,a^2\,b^2\,x^{11}\,\left (A\,b+B\,a\right )}{11}+\frac {5\,a^3\,b\,x^9\,\left (2\,A\,b+B\,a\right )}{9}+\frac {5\,a\,b^3\,x^{13}\,\left (A\,b+2\,B\,a\right )}{13} \]
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