Integrand size = 27, antiderivative size = 12 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]
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Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {24} \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]
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Rule 24
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (b^2 c+b^2 d x\right ) \, dx}{b^2} \\ & = c x+\frac {d x^2}{2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]
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Time = 2.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(\frac {x \left (d x +2 c \right )}{2}\) | \(11\) |
default | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
norman | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
risch | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
parallelrisch | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
parts | \(c x +\frac {1}{2} d \,x^{2}\) | \(11\) |
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none
Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x + \frac {d x^{2}}{2} \]
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none
Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {d\,x^2}{2}+c\,x \]
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