Optimal. Leaf size=88 \[ \frac {(1-2 x)^{3/2}}{63 (3 x+2)^3}+\frac {17 \sqrt {1-2 x}}{441 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{63 (3 x+2)^2}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
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Rubi [A] time = 0.02, 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 = {78, 47, 51, 63, 206} \[ \frac {(1-2 x)^{3/2}}{63 (3 x+2)^3}+\frac {17 \sqrt {1-2 x}}{441 (3 x+2)}-\frac {17 \sqrt {1-2 x}}{63 (3 x+2)^2}+\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{441 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 78
Rule 206
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} \operatorname {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 [C] time = 0.02, size = 42, normalized size = 0.48 \[ \frac {(1-2 x)^{3/2} \left (\frac {343}{(3 x+2)^3}-272 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{21609} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 85, normalized size = 0.97 \[ \frac {17 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (153 \, x^{2} - 167 \, x - 163\right )} \sqrt {-2 \, x + 1}}{9261 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 84, normalized size = 0.95 \[ -\frac {17}{9261} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {153 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}}{1764 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.65 \[ \frac {34 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{9261}+\frac {-\frac {34 \left (-2 x +1\right )^{\frac {5}{2}}}{49}-\frac {8 \left (-2 x +1\right )^{\frac {3}{2}}}{63}+\frac {34 \sqrt {-2 x +1}}{9}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 92, normalized size = 1.05 \[ -\frac {17}{9261} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (153 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 28 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 833 \, \sqrt {-2 \, x + 1}\right )}}{441 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
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 \[ \frac {34\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9261}+\frac {\frac {8\,{\left (1-2\,x\right )}^{3/2}}{1701}-\frac {34\,\sqrt {1-2\,x}}{243}+\frac {34\,{\left (1-2\,x\right )}^{5/2}}{1323}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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