Optimal. Leaf size=68 \[ -\frac {69 \sqrt {1-2 x}}{1210 (5 x+3)}-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}-\frac {69 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]
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Rubi [A] time = 0.01, antiderivative size = 68, 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 = {78, 51, 63, 206} \[ -\frac {69 \sqrt {1-2 x}}{1210 (5 x+3)}-\frac {\sqrt {1-2 x}}{110 (5 x+3)^2}-\frac {69 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x}}{110 (3+5 x)^2}+\frac {69}{110} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x}}{110 (3+5 x)^2}-\frac {69 \sqrt {1-2 x}}{1210 (3+5 x)}+\frac {69 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1210}\\ &=-\frac {\sqrt {1-2 x}}{110 (3+5 x)^2}-\frac {69 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {69 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1210}\\ &=-\frac {\sqrt {1-2 x}}{110 (3+5 x)^2}-\frac {69 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {69 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 0.91 \[ \frac {\sqrt {1-2 x} \left (-\frac {55 (345 x+218)}{(5 x+3)^2}-\frac {138 \sqrt {55} \tan ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right )}{66550} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 69, normalized size = 1.01 \[ \frac {69 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (345 \, x + 218\right )} \sqrt {-2 \, x + 1}}{66550 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 68, normalized size = 1.00 \[ \frac {69}{66550} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 781 \, \sqrt {-2 \, x + 1}}{2420 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.71 \[ -\frac {69 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{33275}-\frac {100 \left (-\frac {69 \left (-2 x +1\right )^{\frac {3}{2}}}{12100}+\frac {71 \sqrt {-2 x +1}}{5500}\right )}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 74, normalized size = 1.09 \[ \frac {69}{66550} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 781 \, \sqrt {-2 \, x + 1}}{605 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 54, normalized size = 0.79 \[ -\frac {69\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275}-\frac {\frac {71\,\sqrt {1-2\,x}}{1375}-\frac {69\,{\left (1-2\,x\right )}^{3/2}}{3025}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]
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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