Integrand size = 22, antiderivative size = 15 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+b x+c x^2} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+b x+c x^2} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {d}{a+b x+c x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+x (b+c x)} \]
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Time = 2.81 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
gosper | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
default | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
risch | \(-\frac {d}{c \,x^{2}+b x +a}\) | \(16\) |
parallelrisch | \(\frac {c d \,x^{2}+b d x}{a \left (c \,x^{2}+b x +a \right )}\) | \(28\) |
norman | \(\frac {\frac {b x d}{a}+\frac {c d \,x^{2}}{a}}{c \,x^{2}+b x +a}\) | \(31\) |
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none
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c x^{2} + b x + a} \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=- \frac {d}{a + b x + c x^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c x^{2} + b x + a} \]
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none
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a + \frac {c d x^{2} + b d x}{d}} \]
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Time = 9.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c\,x^2+b\,x+a} \]
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