Integrand size = 13, antiderivative size = 28 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=\frac {(a+b x)^2}{2 (b c-a d) (c+d x)^2} \]
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Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {37} \[ \int \frac {a+b x}{(c+d x)^3} \, dx=\frac {(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^2}{2 (b c-a d) (c+d x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=-\frac {a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 b d x +a d +b c}{2 d^{2} \left (d x +c \right )^{2}}\) | \(25\) |
parallelrisch | \(\frac {-2 b d x -a d -b c}{2 d^{2} \left (d x +c \right )^{2}}\) | \(27\) |
norman | \(\frac {-\frac {b x}{d}-\frac {a d +b c}{2 d^{2}}}{\left (d x +c \right )^{2}}\) | \(29\) |
risch | \(\frac {-\frac {b x}{d}-\frac {a d +b c}{2 d^{2}}}{\left (d x +c \right )^{2}}\) | \(29\) |
default | \(-\frac {a d -b c}{2 d^{2} \left (d x +c \right )^{2}}-\frac {b}{d^{2} \left (d x +c \right )}\) | \(35\) |
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none
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=\frac {- a d - b c - 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (d x + c\right )}^{2} d^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {a+b x}{(c+d x)^3} \, dx=-\frac {\frac {a\,d+b\,c}{2\,d^2}+\frac {b\,x}{d}}{c^2+2\,c\,d\,x+d^2\,x^2} \]
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