Integrand size = 11, antiderivative size = 24 \[ \int (d x+c (a+b x)) \, dx=\frac {d x^2}{2}+\frac {c (a+b x)^2}{2 b} \]
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Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d x+c (a+b x)) \, dx=\frac {c (a+b x)^2}{2 b}+\frac {d x^2}{2} \]
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Rubi steps \begin{align*} \text {integral}& = \frac {d x^2}{2}+\frac {c (a+b x)^2}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int (d x+c (a+b x)) \, dx=a c x+\frac {1}{2} b c x^2+\frac {d x^2}{2} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {x \left (b c x +2 a c +d x \right )}{2}\) | \(16\) |
norman | \(\left (\frac {b c}{2}+\frac {d}{2}\right ) x^{2}+a c x\) | \(18\) |
default | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
risch | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
parallelrisch | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
parts | \(\frac {1}{2} b c \,x^{2}+a c x +\frac {1}{2} d \,x^{2}\) | \(19\) |
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int (d x+c (a+b x)) \, dx=\frac {1}{2} x^{2} c b + \frac {1}{2} x^{2} d + x c a \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int (d x+c (a+b x)) \, dx=a c x + x^{2} \left (\frac {b c}{2} + \frac {d}{2}\right ) \]
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none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int (d x+c (a+b x)) \, dx=\frac {1}{2} \, d x^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int (d x+c (a+b x)) \, dx=\frac {1}{2} \, d x^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int (d x+c (a+b x)) \, dx=\left (\frac {d}{2}+\frac {b\,c}{2}\right )\,x^2+a\,c\,x \]
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