Optimal. Leaf size=88 \[ \frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65,
212} \begin {gather*} \frac {(1-2 x)^{3/2}}{63 (3 x+2)^3}+\frac {17 \sqrt {1-2 x}}{441 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{63 (3 x+2)^2}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^4} \, dx &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}+\frac {34}{21} \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}-\frac {17}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}-\frac {17}{441} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {17}{441} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {(1-2 x)^{3/2}}{63 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{63 (2+3 x)^2}+\frac {17 \sqrt {1-2 x}}{441 (2+3 x)}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 58, normalized size = 0.66 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-163-167 x+153 x^2\right )}{(2+3 x)^3}+34 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 57, normalized size = 0.65
method | result | size |
risch | \(-\frac {306 x^{3}-487 x^{2}-159 x +163}{441 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(51\) |
derivativedivides | \(\frac {-\frac {34 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{63}+\frac {34 \sqrt {1-2 x}}{9}}{\left (-4-6 x \right )^{3}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(57\) |
default | \(\frac {-\frac {34 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{63}+\frac {34 \sqrt {1-2 x}}{9}}{\left (-4-6 x \right )^{3}}+\frac {34 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9261}\) | \(57\) |
trager | \(\frac {\left (153 x^{2}-167 x -163\right ) \sqrt {1-2 x}}{441 \left (2+3 x \right )^{3}}-\frac {17 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{9261}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.91, size = 92, normalized size = 1.05 \begin {gather*} -\frac {17}{9261} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (153 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}\right )}}{441 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 85, normalized size = 0.97 \begin {gather*} \frac {17 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (153 \, x^{2} - 167 \, x - 163\right )} \sqrt {-2 \, x + 1}}{9261 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 228.26, size = 500, normalized size = 5.68 \begin {gather*} \frac {40 \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 {296 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {112 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 84, normalized size = 0.95 \begin {gather*} -\frac {17}{9261} \, \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 {153 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}}{1764 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 71, normalized size = 0.81 \begin {gather*} \frac {34\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}+\frac {\frac {8\,{\left (1-2\,x\right )}^{3/2}}{1701}-\frac {34\,\sqrt {1-2\,x}}{243}+\frac {34\,{\left (1-2\,x\right )}^{5/2}}{1323}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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