Integrand size = 18, antiderivative size = 42 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {(A b-a B) \left (a+b x^2\right )^3}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {455, 45} \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {\left (a+b x^2\right )^3 (A b-a B)}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (A+B x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^2}{b}+\frac {B (a+b x)^3}{b}\right ) \, dx,x,x^2\right ) \\ & = \frac {(A b-a B) \left (a+b x^2\right )^3}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{24} x^2 \left (12 a^2 A+6 a (2 A b+a B) x^2+4 b (A b+2 a B) x^4+3 b^2 B x^6\right ) \]
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Time = 2.58 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {b^{2} B \,x^{8}}{8}+\frac {\left (b^{2} A +2 a b B \right ) x^{6}}{6}+\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4}+\frac {a^{2} A \,x^{2}}{2}\) | \(52\) |
norman | \(\frac {b^{2} B \,x^{8}}{8}+\left (\frac {1}{6} b^{2} A +\frac {1}{3} a b B \right ) x^{6}+\left (\frac {1}{2} a b A +\frac {1}{4} a^{2} B \right ) x^{4}+\frac {a^{2} A \,x^{2}}{2}\) | \(52\) |
gosper | \(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{6} x^{6} b^{2} A +\frac {1}{3} x^{6} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{2} a^{2} A \,x^{2}\) | \(54\) |
risch | \(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{6} x^{6} b^{2} A +\frac {1}{3} x^{6} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{2} a^{2} A \,x^{2}\) | \(54\) |
parallelrisch | \(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{6} x^{6} b^{2} A +\frac {1}{3} x^{6} a b B +\frac {1}{2} x^{4} a b A +\frac {1}{4} x^{4} a^{2} B +\frac {1}{2} a^{2} A \,x^{2}\) | \(54\) |
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {A a^{2} x^{2}}{2} + \frac {B b^{2} x^{8}}{8} + x^{6} \left (\frac {A b^{2}}{6} + \frac {B a b}{3}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {1}{8} \, B b^{2} x^{8} + \frac {1}{3} \, B a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{2} \, A a^{2} x^{2} \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^6\,\left (\frac {A\,b^2}{6}+\frac {B\,a\,b}{3}\right )+\frac {A\,a^2\,x^2}{2}+\frac {B\,b^2\,x^8}{8} \]
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