Optimal. Leaf size=108 \[ \frac{277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac{14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (3 x+2)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]
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Rubi [A] time = 0.0299983, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{277 (1-2 x)^{5/2}}{5292 (3 x+2)^3}-\frac{(1-2 x)^{5/2}}{252 (3 x+2)^4}-\frac{14423 (1-2 x)^{3/2}}{31752 (3 x+2)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (3 x+2)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{1}{252} \int \frac{(1-2 x)^{3/2} (1123+2100 x)}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}+\frac{14423 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac{14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac{14423 \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac{14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (2+3 x)}+\frac{14423 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{31752}\\ &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac{14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (2+3 x)}-\frac{14423 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{31752}\\ &=-\frac{(1-2 x)^{5/2}}{252 (2+3 x)^4}+\frac{277 (1-2 x)^{5/2}}{5292 (2+3 x)^3}-\frac{14423 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac{14423 \sqrt{1-2 x}}{31752 (2+3 x)}-\frac{14423 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0591249, size = 79, normalized size = 0.73 \[ \frac{-21 \left (1337958 x^4+1307091 x^3-80575 x^2-331950 x-60890\right )-28846 \sqrt{21-42 x} (3 x+2)^4 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{666792 \sqrt{1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.6 \begin{align*} 648\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{2753\, \left ( 1-2\,x \right ) ^{7/2}}{42336}}+{\frac{189667\, \left ( 1-2\,x \right ) ^{5/2}}{489888}}-{\frac{158653\, \left ( 1-2\,x \right ) ^{3/2}}{209952}}+{\frac{100961\,\sqrt{1-2\,x}}{209952}} \right ) }-{\frac{14423\,\sqrt{21}}{333396}{\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 = 2.42277, size = 149, normalized size = 1.38 \begin{align*} \frac{14423}{666792} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{668979 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 3983007 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 7773997 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 4947089 \, \sqrt{-2 \, x + 1}}{15876 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40141, size = 304, normalized size = 2.81 \begin{align*} \frac{14423 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (668979 \, x^{3} + 988035 \, x^{2} + 453730 \, x + 60890\right )} \sqrt{-2 \, x + 1}}{666792 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 2.30889, size = 135, normalized size = 1.25 \begin{align*} \frac{14423}{666792} \, \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{668979 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 3983007 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 7773997 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 4947089 \, \sqrt{-2 \, x + 1}}{254016 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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