Optimal. Leaf size=88 \[ -\frac {50 \sqrt {1-2 x}}{1029 (3 x+2)}-\frac {50 \sqrt {1-2 x}}{441 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}-\frac {100 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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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 = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {50 \sqrt {1-2 x}}{1029 (3 x+2)}-\frac {50 \sqrt {1-2 x}}{441 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{63 (3 x+2)^3}-\frac {100 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx &=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}+\frac {100}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}-\frac {50 \sqrt {1-2 x}}{441 (2+3 x)^2}+\frac {50}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}-\frac {50 \sqrt {1-2 x}}{441 (2+3 x)^2}-\frac {50 \sqrt {1-2 x}}{1029 (2+3 x)}+\frac {50 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1029}\\ &=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}-\frac {50 \sqrt {1-2 x}}{441 (2+3 x)^2}-\frac {50 \sqrt {1-2 x}}{1029 (2+3 x)}-\frac {50 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1029}\\ &=\frac {\sqrt {1-2 x}}{63 (2+3 x)^3}-\frac {50 \sqrt {1-2 x}}{441 (2+3 x)^2}-\frac {50 \sqrt {1-2 x}}{1029 (2+3 x)}-\frac {100 \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 = 0.48 \[ \frac {\sqrt {1-2 x} \left (\frac {343}{(3 x+2)^3}-800 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{21609} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 84, normalized size = 0.95 \[ \frac {50 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (450 \, x^{2} + 950 \, x + 417\right )} \sqrt {-2 \, x + 1}}{21609 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 84, normalized size = 0.95 \[ \frac {50}{21609} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {225 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1400 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2009 \, \sqrt {-2 \, x + 1}}{2058 \, {\left (3 \, x + 2\right )}^{3}} \]
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 \[ -\frac {100 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}+\frac {\frac {300 \left (-2 x +1\right )^{\frac {5}{2}}}{343}-\frac {800 \left (-2 x +1\right )^{\frac {3}{2}}}{147}+\frac {164 \sqrt {-2 x +1}}{21}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 92, normalized size = 1.05 \[ \frac {50}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (225 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1400 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2009 \, \sqrt {-2 \, x + 1}\right )}}{1029 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 72, normalized size = 0.82 \[ -\frac {100\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-\frac {\frac {164\,\sqrt {1-2\,x}}{567}-\frac {800\,{\left (1-2\,x\right )}^{3/2}}{3969}+\frac {100\,{\left (1-2\,x\right )}^{5/2}}{3087}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]
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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