Optimal. Leaf size=31 \[ \frac {3}{5 (4+5 x)}+\frac {\sqrt {1-x^2}}{4+5 x} \]
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Rubi [A]
time = 0.19, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6874, 745, 739,
212, 821} \begin {gather*} \frac {\sqrt {1-x^2}}{5 x+4}+\frac {3}{5 (5 x+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 745
Rule 821
Rule 6874
Rubi steps
\begin {align*} \int \frac {-5-4 x-3 \sqrt {1-x^2}}{(4+5 x)^2 \sqrt {1-x^2}} \, dx &=\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*}
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Mathematica [A]
time = 0.26, size = 23, normalized size = 0.74 \begin {gather*} \frac {3+5 \sqrt {1-x^2}}{20+25 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 32, normalized size = 1.03
method | result | size |
trager | \(-\frac {3 x}{4 \left (5 x +4\right )}+\frac {\sqrt {-x^{2}+1}}{5 x +4}\) | \(29\) |
default | \(\frac {\sqrt {-\left (x +\frac {4}{5}\right )^{2}+\frac {8 x}{5}+\frac {41}{25}}}{5 x +4}+\frac {3}{5 \left (5 x +4\right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 25, normalized size = 0.81 \begin {gather*} \frac {5 \, \sqrt {x + 1} \sqrt {-x + 1} + 3}{5 \, {\left (5 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 0.81 \begin {gather*} \frac {25 \, x + 20 \, \sqrt {-x^{2} + 1} + 32}{20 \, {\left (5 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.08, size = 54, normalized size = 1.74 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 19, normalized size = 0.61 \begin {gather*} \frac {\sqrt {1-x^2}+\frac {3}{5}}{5\,x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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