Integrand size = 27, antiderivative size = 28 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 37} \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
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Rule 24
Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^3} \, dx}{b^2} \\ & = -\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 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]
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Time = 2.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(-\frac {2 b d x +a d +b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(25\) |
parallelrisch | \(\frac {-2 b d x -a d -b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(27\) |
risch | \(\frac {-\frac {d x}{b}-\frac {a d +b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
default | \(-\frac {-a d +b c}{2 b^{2} \left (b x +a \right )^{2}}-\frac {d}{b^{2} \left (b x +a \right )}\) | \(35\) |
norman | \(\frac {-d \,x^{2}+\frac {a \left (-a b d -b^{2} c \right )}{2 b^{3}}+\frac {\left (-3 a b d -b^{2} c \right ) x}{2 b^{2}}}{\left (b x +a \right )^{3}}\) | \(52\) |
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none
Time = 0.43 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=\frac {- a d - b c - 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]
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none
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {2 \, b d x + b c + a d}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
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Time = 9.84 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx=-\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
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