Integrand size = 13, antiderivative size = 28 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {(A+B x)^2}{2 (A b-a B) (a+b x)^2} \]
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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} \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {(A+B x)^2}{2 (a+b x)^2 (A b-a B)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+B x)^2}{2 (A b-a B) (a+b x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {A b+B (a+2 b x)}{2 b^2 (a+b x)^2} \]
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Time = 2.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 b B x +A b +B a}{2 b^{2} \left (b x +a \right )^{2}}\) | \(25\) |
parallelrisch | \(-\frac {2 b B x +A b +B a}{2 b^{2} \left (b x +a \right )^{2}}\) | \(25\) |
norman | \(\frac {-\frac {B x}{b}-\frac {A b +B a}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
risch | \(\frac {-\frac {B x}{b}-\frac {A b +B a}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
default | \(-\frac {A b -B a}{2 b^{2} \left (b x +a \right )^{2}}-\frac {B}{b^{2} \left (b x +a \right )}\) | \(35\) |
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none
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {2 \, B b x + B a + A b}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=\frac {- A b - B a - 2 B b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {2 \, B b x + B a + A b}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {2 \, B b x + B a + A b}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {A+B x}{(a+b x)^3} \, dx=-\frac {\frac {A\,b+B\,a}{2\,b^2}+\frac {B\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
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