Optimal. Leaf size=128 \[ \frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x}}{1029 (3 x+2)}+\frac {\sqrt {1-2 x}}{441 (3 x+2)^2}+\frac {2 \sqrt {1-2 x}}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x}}{15 (3 x+2)^4}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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Rubi [A] time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x}}{1029 (3 x+2)}+\frac {\sqrt {1-2 x}}{441 (3 x+2)^2}+\frac {2 \sqrt {1-2 x}}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x}}{15 (3 x+2)^4}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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)^6} \, dx &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}+\frac {8}{5} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}-\frac {2}{15} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}-\frac {2}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\sqrt {1-2 x}}{441 (2+3 x)^2}-\frac {1}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\sqrt {1-2 x}}{441 (2+3 x)^2}+\frac {\sqrt {1-2 x}}{1029 (2+3 x)}-\frac {\int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1029}\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\sqrt {1-2 x}}{441 (2+3 x)^2}+\frac {\sqrt {1-2 x}}{1029 (2+3 x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1029}\\ &=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\sqrt {1-2 x}}{441 (2+3 x)^2}+\frac {\sqrt {1-2 x}}{1029 (2+3 x)}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.33 \[ \frac {(1-2 x)^{3/2} \left (\frac {2401}{(3 x+2)^5}-256 \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{252105} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 115, normalized size = 0.90 \[ \frac {5 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (405 \, x^{4} + 1395 \, x^{3} + 2004 \, x^{2} - 864 \, x - 1019\right )} \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 116, normalized size = 0.91 \[ -\frac {1}{21609} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {405 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 4410 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 18816 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 19110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12005 \, \sqrt {-2 \, x + 1}}{82320 \, {\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.59 \[ \frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}+\frac {-\frac {54 \left (-2 x +1\right )^{\frac {9}{2}}}{343}+\frac {12 \left (-2 x +1\right )^{\frac {7}{2}}}{7}-\frac {256 \left (-2 x +1\right )^{\frac {5}{2}}}{35}+\frac {52 \left (-2 x +1\right )^{\frac {3}{2}}}{7}+\frac {14 \sqrt {-2 x +1}}{3}}{\left (-6 x -4\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 128, normalized size = 1.00 \[ -\frac {1}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (405 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 4410 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 18816 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 19110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12005 \, \sqrt {-2 \, x + 1}\right )}}{5145 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 108, normalized size = 0.84 \[ \frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-\frac {\frac {14\,\sqrt {1-2\,x}}{729}+\frac {52\,{\left (1-2\,x\right )}^{3/2}}{1701}-\frac {256\,{\left (1-2\,x\right )}^{5/2}}{8505}+\frac {4\,{\left (1-2\,x\right )}^{7/2}}{567}-\frac {2\,{\left (1-2\,x\right )}^{9/2}}{3087}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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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