Integrand size = 28, antiderivative size = 14 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x+\frac {1}{2} c e x^2 \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {24} \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x+\frac {1}{2} c e x^2 \]
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Rule 24
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (c d e^2+c e^3 x\right ) \, dx}{e^2} \\ & = c d x+\frac {1}{2} c e x^2 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c \left (d x+\frac {e x^2}{2}\right ) \]
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Time = 2.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) c}{2}\) | \(12\) |
default | \(c \left (\frac {1}{2} e \,x^{2}+d x \right )\) | \(13\) |
norman | \(c d x +\frac {1}{2} c e \,x^{2}\) | \(13\) |
risch | \(c d x +\frac {1}{2} c e \,x^{2}\) | \(13\) |
parallelrisch | \(c d x +\frac {1}{2} c e \,x^{2}\) | \(13\) |
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Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=c d x + \frac {c e x^{2}}{2} \]
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Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]
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Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {1}{2} \, c e x^{2} + c d x \]
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Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx=\frac {c\,x\,\left (2\,d+e\,x\right )}{2} \]
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