Integrand size = 16, antiderivative size = 37 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} A c x^5+\frac {1}{6} B c x^6 \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {780} \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} A c x^5+\frac {1}{6} B c x^6 \]
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Rule 780
Rubi steps \begin{align*} \text {integral}& = \int \left (a A x^2+a B x^3+A c x^4+B c x^5\right ) \, dx \\ & = \frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} A c x^5+\frac {1}{6} B c x^6 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} A c x^5+\frac {1}{6} B c x^6 \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {1}{3} a A \,x^{3}+\frac {1}{4} a B \,x^{4}+\frac {1}{5} A c \,x^{5}+\frac {1}{6} B c \,x^{6}\) | \(30\) |
default | \(\frac {1}{3} a A \,x^{3}+\frac {1}{4} a B \,x^{4}+\frac {1}{5} A c \,x^{5}+\frac {1}{6} B c \,x^{6}\) | \(30\) |
norman | \(\frac {1}{3} a A \,x^{3}+\frac {1}{4} a B \,x^{4}+\frac {1}{5} A c \,x^{5}+\frac {1}{6} B c \,x^{6}\) | \(30\) |
risch | \(\frac {1}{3} a A \,x^{3}+\frac {1}{4} a B \,x^{4}+\frac {1}{5} A c \,x^{5}+\frac {1}{6} B c \,x^{6}\) | \(30\) |
parallelrisch | \(\frac {1}{3} a A \,x^{3}+\frac {1}{4} a B \,x^{4}+\frac {1}{5} A c \,x^{5}+\frac {1}{6} B c \,x^{6}\) | \(30\) |
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none
Time = 0.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c x^{6} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, B a x^{4} + \frac {1}{3} \, A a x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {A a x^{3}}{3} + \frac {A c x^{5}}{5} + \frac {B a x^{4}}{4} + \frac {B c x^{6}}{6} \]
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none
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c x^{6} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, B a x^{4} + \frac {1}{3} \, A a x^{3} \]
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none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {1}{6} \, B c x^{6} + \frac {1}{5} \, A c x^{5} + \frac {1}{4} \, B a x^{4} + \frac {1}{3} \, A a x^{3} \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x^2 (A+B x) \left (a+c x^2\right ) \, dx=\frac {B\,c\,x^6}{6}+\frac {A\,c\,x^5}{5}+\frac {B\,a\,x^4}{4}+\frac {A\,a\,x^3}{3} \]
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