Optimal. Leaf size=88 \[ \frac{23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac{(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac{2381 \sqrt{1-2 x}}{2646 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1323 \sqrt{21}} \]
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Rubi [A] time = 0.0214066, antiderivative size = 88, 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 = {89, 78, 47, 63, 206} \[ \frac{23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac{(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac{2381 \sqrt{1-2 x}}{2646 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1323 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac{1}{189} \int \frac{\sqrt{1-2 x} (843+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac{23 (1-2 x)^{3/2}}{294 (2+3 x)^2}+\frac{2381}{882} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac{23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac{2381 \sqrt{1-2 x}}{2646 (2+3 x)}-\frac{2381 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2646}\\ &=-\frac{(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac{23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac{2381 \sqrt{1-2 x}}{2646 (2+3 x)}+\frac{2381 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2646}\\ &=-\frac{(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac{23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac{2381 \sqrt{1-2 x}}{2646 (2+3 x)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1323 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0483531, size = 74, normalized size = 0.84 \[ \frac{21 \left (45342 x^3+34831 x^2-10503 x-9124\right )+4762 \sqrt{21-42 x} (3 x+2)^3 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 57, normalized size = 0.7 \begin{align*} -108\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{2519\, \left ( 1-2\,x \right ) ^{5/2}}{15876}}+{\frac{3673\, \left ( 1-2\,x \right ) ^{3/2}}{5103}}-{\frac{2381\,\sqrt{1-2\,x}}{2916}} \right ) }+{\frac{2381\,\sqrt{21}}{27783}{\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 = 2.69325, size = 124, normalized size = 1.41 \begin{align*} -\frac{2381}{55566} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{22671 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 102844 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 116669 \, \sqrt{-2 \, x + 1}}{1323 \,{\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 = 1.59673, size = 247, normalized size = 2.81 \begin{align*} \frac{2381 \, \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 (22671 \, x^{2} + 28751 \, x + 9124\right )} \sqrt{-2 \, x + 1}}{55566 \,{\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 = 1.91003, size = 113, normalized size = 1.28 \begin{align*} -\frac{2381}{55566} \, \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{22671 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 102844 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 116669 \, \sqrt{-2 \, x + 1}}{10584 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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