Optimal. Leaf size=107 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac{2 \sqrt{1-2 x} (26075 x+18016)}{3969 (3 x+2)}-\frac{92996 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]
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Rubi [A] time = 0.0281685, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {97, 149, 146, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac{2 \sqrt{1-2 x} (26075 x+18016)}{3969 (3 x+2)}-\frac{92996 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 146
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac{1}{378} \int \frac{(544-2980 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (18016+26075 x)}{3969 (2+3 x)}+\frac{46498 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{3969}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac{46498 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3969}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac{2 \sqrt{1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac{92996 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0498399, size = 79, normalized size = 0.74 \[ \frac{-21 \left (661500 x^4+1059336 x^3+274193 x^2-260244 x-112187\right )-92996 \sqrt{21-42 x} (3 x+2)^3 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{83349 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 66, normalized size = 0.6 \begin{align*}{\frac{250}{81}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3727}{147} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{22046}{189} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3623}{27}\sqrt{1-2\,x}} \right ) }-{\frac{92996\,\sqrt{21}}{83349}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.68754, size = 136, normalized size = 1.27 \begin{align*} \frac{46498}{83349} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{250}{81} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (33543 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 154322 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177527 \, \sqrt{-2 \, x + 1}\right )}}{3969 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6118, size = 271, normalized size = 2.53 \begin{align*} \frac{46498 \, \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 (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt{-2 \, x + 1}}{83349 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.64686, size = 126, normalized size = 1.18 \begin{align*} \frac{46498}{83349} \, \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{250}{81} \, \sqrt{-2 \, x + 1} + \frac{33543 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 154322 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177527 \, \sqrt{-2 \, x + 1}}{15876 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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