Optimal. Leaf size=27 \[ \frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x} \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {5}{2} \sqrt {1-2 x}+\frac {11}{2 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {11}{2 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}}\right ) \, dx\\ &=\frac {11}{2 \sqrt {1-2 x}}+\frac {5}{2} \sqrt {1-2 x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 15, normalized size = 0.56 \begin {gather*} \frac {8-5 x}{\sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 0.74
method | result | size |
gosper | \(-\frac {-8+5 x}{\sqrt {1-2 x}}\) | \(15\) |
risch | \(-\frac {-8+5 x}{\sqrt {1-2 x}}\) | \(15\) |
derivativedivides | \(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\) | \(20\) |
default | \(\frac {11}{2 \sqrt {1-2 x}}+\frac {5 \sqrt {1-2 x}}{2}\) | \(20\) |
trager | \(\frac {\left (-8+5 x \right ) \sqrt {1-2 x}}{-1+2 x}\) | \(21\) |
meijerg | \(-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-5 \sqrt {\pi }+\frac {5 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 20, normalized size = 0.74 \begin {gather*} \frac {{\left (5 \, x - 8\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 31, normalized size = 1.15 \begin {gather*} \frac {5 x \sqrt {1 - 2 x}}{2 x - 1} - \frac {8 \sqrt {1 - 2 x}}{2 x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.28, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{2} \, \sqrt {-2 \, x + 1} + \frac {11}{2 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.52 \begin {gather*} -\frac {5\,x-8}{\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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