Optimal. Leaf size=27 \[ -\frac {11}{2} \sqrt {1-2 x}+\frac {5}{6} (1-2 x)^{3/2} \]
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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}{6} (1-2 x)^{3/2}-\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}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {11}{2 \sqrt {1-2 x}}-\frac {5}{2} \sqrt {1-2 x}\right ) \, dx\\ &=-\frac {11}{2} \sqrt {1-2 x}+\frac {5}{6} (1-2 x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3} \sqrt {1-2 x} (14+5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 20, normalized size = 0.74
method | result | size |
trager | \(\left (-\frac {14}{3}-\frac {5 x}{3}\right ) \sqrt {1-2 x}\) | \(14\) |
gosper | \(-\frac {\left (14+5 x \right ) \sqrt {1-2 x}}{3}\) | \(15\) |
derivativedivides | \(\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{6}-\frac {11 \sqrt {1-2 x}}{2}\) | \(20\) |
default | \(\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{6}-\frac {11 \sqrt {1-2 x}}{2}\) | \(20\) |
risch | \(\frac {\left (-1+2 x \right ) \left (14+5 x \right )}{3 \sqrt {1-2 x}}\) | \(20\) |
meijerg | \(-\frac {3 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{2 \sqrt {\pi }}+\frac {\frac {5 \sqrt {\pi }}{3}-\frac {5 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{24}}{\sqrt {\pi }}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \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 = 0.96, size = 14, normalized size = 0.52 \begin {gather*} -\frac {1}{3} \, {\left (5 \, x + 14\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 87, normalized size = 3.22 \begin {gather*} \begin {cases} - \frac {\sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{3} - \frac {11 \sqrt {5} i \sqrt {10 x - 5}}{15} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {\sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{3} - \frac {11 \sqrt {5} \sqrt {5 - 10 x}}{15} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \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 = 0.02, size = 14, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {1-2\,x}\,\left (10\,x+28\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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