Integrand size = 20, antiderivative size = 34 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {621} \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
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Rule 621
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {a+b x}{4 b \left ((a+b x)^2\right )^{5/2}} \]
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Time = 2.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {b x +a}{4 b \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(20\) |
default | \(-\frac {b x +a}{4 b \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(20\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}}{4 \left (b x +a \right )^{5} b}\) | \(22\) |
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none
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]
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\[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {1}{4 \, {\left (b x + a\right )}^{4} b \mathrm {sgn}\left (b x + a\right )} \]
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Time = 9.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b\,{\left (a+b\,x\right )}^5} \]
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