Integrand size = 13, antiderivative size = 28 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2} \]
[Out]
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} \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
[In]
[Out]
Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 b d x +a d +b c}{2 \left (b x +a \right )^{2} b^{2}}\) | \(25\) |
parallelrisch | \(\frac {-2 b d x -a d -b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(27\) |
risch | \(\frac {-\frac {d x}{b}-\frac {a d +b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
norman | \(\frac {-\frac {d x}{b}+\frac {-a d -b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(31\) |
default | \(-\frac {-a d +b c}{2 b^{2} \left (b x +a \right )^{2}}-\frac {d}{b^{2} \left (b x +a \right )}\) | \(35\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=\frac {- a d - b c - 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {c+d x}{(a+b x)^3} \, dx=-\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
[In]
[Out]