Integrand size = 20, antiderivative size = 55 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{5} a (2 A b+a B) x^5+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{9} b^2 B x^9 \]
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Time = 0.03 (sec) , antiderivative size = 55, 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} \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{9} b^2 B x^9 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A x^2+a (2 A b+a B) x^4+b (A b+2 a B) x^6+b^2 B x^8\right ) \, dx \\ & = \frac {1}{3} a^2 A x^3+\frac {1}{5} a (2 A b+a B) x^5+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{9} b^2 B x^9 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{3} a^2 A x^3+\frac {1}{5} a (2 A b+a B) x^5+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{9} b^2 B x^9 \]
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Time = 2.48 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {b^{2} B \,x^{9}}{9}+\frac {\left (b^{2} A +2 a b B \right ) x^{7}}{7}+\frac {\left (2 a b A +a^{2} B \right ) x^{5}}{5}+\frac {a^{2} A \,x^{3}}{3}\) | \(52\) |
norman | \(\frac {b^{2} B \,x^{9}}{9}+\left (\frac {1}{7} b^{2} A +\frac {2}{7} a b B \right ) x^{7}+\left (\frac {2}{5} a b A +\frac {1}{5} a^{2} B \right ) x^{5}+\frac {a^{2} A \,x^{3}}{3}\) | \(52\) |
gosper | \(\frac {1}{9} b^{2} B \,x^{9}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {2}{5} x^{5} a b A +\frac {1}{5} x^{5} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
risch | \(\frac {1}{9} b^{2} B \,x^{9}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {2}{5} x^{5} a b A +\frac {1}{5} x^{5} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
parallelrisch | \(\frac {1}{9} b^{2} B \,x^{9}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {2}{5} x^{5} a b A +\frac {1}{5} x^{5} a^{2} B +\frac {1}{3} a^{2} A \,x^{3}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{9} \, B b^{2} x^{9} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{5} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {A a^{2} x^{3}}{3} + \frac {B b^{2} x^{9}}{9} + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7}\right ) + x^{5} \cdot \left (\frac {2 A a b}{5} + \frac {B a^{2}}{5}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{9} \, B b^{2} x^{9} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{5} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{9} \, B b^{2} x^{9} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{5} \, B a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{3} \, A a^{2} x^{3} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x^5\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,b^2\,x^9}{9} \]
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