Optimal. Leaf size=59 \[ \frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 52, 65, 212}
\begin {gather*} \frac {(1-2 x)^{3/2}}{21 (3 x+2)}+\frac {8}{7} \sqrt {1-2 x}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^2} \, dx &=\frac {(1-2 x)^{3/2}}{21 (2+3 x)}+\frac {12}{7} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}+4 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}-4 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {8}{7} \sqrt {1-2 x}+\frac {(1-2 x)^{3/2}}{21 (2+3 x)}-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 55, normalized size = 0.93 \begin {gather*} \frac {7 \sqrt {1-2 x} (7+10 x)-8 \sqrt {21} (2+3 x) \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{42+63 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 45, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {10 \sqrt {1-2 x}}{9}-\frac {2 \sqrt {1-2 x}}{27 \left (-\frac {4}{3}-2 x \right )}-\frac {8 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(45\) |
default | \(\frac {10 \sqrt {1-2 x}}{9}-\frac {2 \sqrt {1-2 x}}{27 \left (-\frac {4}{3}-2 x \right )}-\frac {8 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(45\) |
risch | \(-\frac {20 x^{2}+4 x -7}{3 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {8 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(46\) |
trager | \(\frac {\left (10 x +7\right ) \sqrt {1-2 x}}{6+9 x}-\frac {4 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{21}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 62, normalized size = 1.05 \begin {gather*} \frac {4}{21} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {10}{9} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 59, normalized size = 1.00 \begin {gather*} \frac {4 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (10 \, x + 7\right )} \sqrt {-2 \, x + 1}}{21 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 53.13, size = 192, normalized size = 3.25 \begin {gather*} \frac {10 \sqrt {1 - 2 x}}{9} + \frac {28 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {74 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 65, normalized size = 1.10 \begin {gather*} \frac {4}{21} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {10}{9} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 44, normalized size = 0.75 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{27\,\left (2\,x+\frac {4}{3}\right )}-\frac {8\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21}+\frac {10\,\sqrt {1-2\,x}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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