Integrand size = 15, antiderivative size = 30 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {b^2 x^9}{9} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 276} \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {b^2 x^9}{9} \]
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Rule 276
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x^4 \left (a+b x^2\right )^2 \, dx \\ & = \int \left (a^2 x^4+2 a b x^6+b^2 x^8\right ) \, dx \\ & = \frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {b^2 x^9}{9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {b^2 x^9}{9} \]
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Time = 1.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {1}{5} x^{5} a^{2}+\frac {2}{7} a b \,x^{7}+\frac {1}{9} b^{2} x^{9}\) | \(25\) |
norman | \(\frac {1}{5} x^{5} a^{2}+\frac {2}{7} a b \,x^{7}+\frac {1}{9} b^{2} x^{9}\) | \(25\) |
risch | \(\frac {1}{5} x^{5} a^{2}+\frac {2}{7} a b \,x^{7}+\frac {1}{9} b^{2} x^{9}\) | \(25\) |
parallelrisch | \(\frac {1}{5} x^{5} a^{2}+\frac {2}{7} a b \,x^{7}+\frac {1}{9} b^{2} x^{9}\) | \(25\) |
gosper | \(\frac {x^{5} \left (35 b^{2} x^{4}+90 a b \,x^{2}+63 a^{2}\right )}{315}\) | \(27\) |
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none
Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {a^{2} x^{5}}{5} + \frac {2 a b x^{7}}{7} + \frac {b^{2} x^{9}}{9} \]
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none
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]
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none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a x+b x^3\right )^2 \, dx=\frac {a^2\,x^5}{5}+\frac {2\,a\,b\,x^7}{7}+\frac {b^2\,x^9}{9} \]
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