Integrand size = 12, antiderivative size = 25 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]
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Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14} \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x^2+c x^3\right ) \, dx \\ & = \frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^2}{2}+\frac {b x^3}{3}+\frac {c x^4}{4} \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\) | \(20\) |
default | \(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\) | \(20\) |
norman | \(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\) | \(20\) |
risch | \(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\) | \(20\) |
parallelrisch | \(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}+\frac {1}{4} c \,x^{4}\) | \(20\) |
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none
Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {a x^{2}}{2} + \frac {b x^{3}}{3} + \frac {c x^{4}}{4} \]
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none
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c x^{4} + \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x \left (a+b x+c x^2\right ) \, dx=\frac {x^2\,\left (3\,c\,x^2+4\,b\,x+6\,a\right )}{12} \]
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