Integrand size = 37, antiderivative size = 31 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6874, 745, 739, 212, 821} \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {\sqrt {1-x^2}}{5 x+4}+\frac {3}{5 (5 x+4)} \]
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Rule 212
Rule 739
Rule 745
Rule 821
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3}{(4+5 x)^2}-\frac {5}{(4+5 x)^2 \sqrt {1-x^2}}-\frac {4 x}{(4+5 x)^2 \sqrt {1-x^2}}\right ) \, dx \\ & = \frac {3}{5 (4+5 x)}-4 \int \frac {x}{(4+5 x)^2 \sqrt {1-x^2}} \, dx-5 \int \frac {1}{(4+5 x)^2 \sqrt {1-x^2}} \, dx \\ & = \frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {3+5 \sqrt {1-x^2}}{20+25 x} \]
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Time = 1.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
trager | \(-\frac {3 x}{4 \left (4+5 x \right )}+\frac {\sqrt {-x^{2}+1}}{4+5 x}\) | \(29\) |
default | \(\frac {\sqrt {-\left (\frac {4}{5}+x \right )^{2}+\frac {8 x}{5}+\frac {41}{25}}}{4+5 x}+\frac {3}{5 \left (4+5 x \right )}\) | \(32\) |
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {25 \, x + 20 \, \sqrt {-x^{2} + 1} + 32}{20 \, {\left (5 \, x + 4\right )}} \]
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\[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=- \int \frac {4 x}{25 x^{2} \sqrt {1 - x^{2}} + 40 x \sqrt {1 - x^{2}} + 16 \sqrt {1 - x^{2}}}\, dx - \int \frac {3 \sqrt {1 - x^{2}}}{25 x^{2} \sqrt {1 - x^{2}} + 40 x \sqrt {1 - x^{2}} + 16 \sqrt {1 - x^{2}}}\, dx - \int \frac {5}{25 x^{2} \sqrt {1 - x^{2}} + 40 x \sqrt {1 - x^{2}} + 16 \sqrt {1 - x^{2}}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {5 \, \sqrt {x + 1} \sqrt {-x + 1} + 3}{5 \, {\left (5 \, x + 4\right )}} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {\sqrt {\frac {8}{5 \, x + 4} + \frac {9}{{\left (5 \, x + 4\right )}^{2}} - 1}}{5 \, \mathrm {sgn}\left (\frac {1}{5 \, x + 4}\right )} + \frac {3}{5 \, {\left (5 \, x + 4\right )}} - \frac {1}{5} i \, \mathrm {sgn}\left (\frac {1}{5 \, x + 4}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx=\frac {\sqrt {1-x^2}+\frac {3}{5}}{5\,x+4} \]
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