Integrand size = 15, antiderivative size = 33 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{2} b d x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4 \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {645} \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{3} x^3 (b e+c d)+\frac {1}{2} b d x^2+\frac {1}{4} c e x^4 \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (b d x+(c d+b e) x^2+c e x^3\right ) \, dx \\ & = \frac {1}{2} b d x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{12} x^2 (c x (4 d+3 e x)+b (6 d+4 e x)) \]
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Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {x^{2} \left (3 c e \,x^{2}+4 b e x +4 c d x +6 b d \right )}{12}\) | \(28\) |
default | \(\frac {b d \,x^{2}}{2}+\frac {\left (b e +c d \right ) x^{3}}{3}+\frac {c e \,x^{4}}{4}\) | \(28\) |
norman | \(\frac {c e \,x^{4}}{4}+\left (\frac {b e}{3}+\frac {c d}{3}\right ) x^{3}+\frac {b d \,x^{2}}{2}\) | \(29\) |
risch | \(\frac {1}{2} b d \,x^{2}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{4} c e \,x^{4}\) | \(30\) |
parallelrisch | \(\frac {1}{2} b d \,x^{2}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{4} c e \,x^{4}\) | \(30\) |
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none
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{2} \, b d x^{2} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {b d x^{2}}{2} + \frac {c e x^{4}}{4} + x^{3} \left (\frac {b e}{3} + \frac {c d}{3}\right ) \]
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none
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{2} \, b d x^{2} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} \]
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none
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{3} \, c d x^{3} + \frac {1}{3} \, b e x^{3} + \frac {1}{2} \, b d x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int (d+e x) \left (b x+c x^2\right ) \, dx=\frac {c\,e\,x^4}{4}+\left (\frac {b\,e}{3}+\frac {c\,d}{3}\right )\,x^3+\frac {b\,d\,x^2}{2} \]
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