Integrand size = 18, antiderivative size = 25 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^4}{4}+\frac {b x^6}{6}+\frac {c x^8}{8} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^4}{4}+\frac {b x^6}{6}+\frac {c x^8}{8} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3+b x^5+c x^7\right ) \, dx \\ & = \frac {a x^4}{4}+\frac {b x^6}{6}+\frac {c x^8}{8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^4}{4}+\frac {b x^6}{6}+\frac {c x^8}{8} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}+\frac {1}{8} c \,x^{8}\) | \(20\) |
norman | \(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}+\frac {1}{8} c \,x^{8}\) | \(20\) |
risch | \(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}+\frac {1}{8} c \,x^{8}\) | \(20\) |
parallelrisch | \(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}+\frac {1}{8} c \,x^{8}\) | \(20\) |
gosper | \(\frac {x^{4} \left (3 c \,x^{4}+4 b \,x^{2}+6 a \right )}{24}\) | \(22\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{8} \, c x^{8} + \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {a x^{4}}{4} + \frac {b x^{6}}{6} + \frac {c x^{8}}{8} \]
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none
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{8} \, c x^{8} + \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \]
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none
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {1}{8} \, c x^{8} + \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a x+b x^3+c x^5\right ) \, dx=\frac {c\,x^8}{8}+\frac {b\,x^6}{6}+\frac {a\,x^4}{4} \]
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