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 = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {817}
\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 817
Rubi steps
\begin {align*} \int \left (-\frac {3}{(4+5 x)^2}-\frac {5+4 x}{(4+5 x)^2 \sqrt {1-x^2}}\right ) \, dx &=\frac {3}{5 (4+5 x)}-\int \frac {5+4 x}{(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 = 0.27, size = 39, normalized size = 1.26 \begin {gather*} \frac {3+\frac {5}{\sqrt {1-x^2}}-\frac {5 x^2}{\sqrt {1-x^2}}}{20+25 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, 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 | \(\frac {\sqrt {-\left (x +\frac {4}{5}\right )^{2}+\frac {8 x}{5}+\frac {41}{25}}}{5 x +4}+\frac {3}{5 \left (5 x +4\right )}\) | \(32\) |
risch | \(\frac {3}{25 \left (x +\frac {4}{5}\right )}-\frac {x^{2}-1}{\left (5 x +4\right ) \sqrt {-x^{2}+1}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 27, normalized size = 0.87 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{5 \, x + 4} + \frac {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 = 0.34, 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 = 2.34, 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 = 3.33, 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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