Optimal. Leaf size=109 \[ \frac{143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac{(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac{211}{441} (1-2 x)^{5/2}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{1055}{81} \sqrt{1-2 x}-\frac{1055}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0341017, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \[ \frac{143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac{(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac{211}{441} (1-2 x)^{5/2}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{1055}{81} \sqrt{1-2 x}-\frac{1055}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{1}{126} \int \frac{(1-2 x)^{5/2} (557+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac{1055}{294} \int \frac{(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac{211}{441} (1-2 x)^{5/2}-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac{1055}{126} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{1055}{567} (1-2 x)^{3/2}+\frac{211}{441} (1-2 x)^{5/2}-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac{1055}{54} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{1055}{81} \sqrt{1-2 x}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{211}{441} (1-2 x)^{5/2}-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac{7385}{162} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1055}{81} \sqrt{1-2 x}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{211}{441} (1-2 x)^{5/2}-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac{7385}{162} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{1055}{81} \sqrt{1-2 x}+\frac{1055}{567} (1-2 x)^{3/2}+\frac{211}{441} (1-2 x)^{5/2}-\frac{(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac{143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac{1055}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0409908, size = 68, normalized size = 0.62 \[ \frac{1}{486} \left (\frac{3 \sqrt{1-2 x} \left (2160 x^4-3960 x^3+12828 x^2+25987 x+10007\right )}{(3 x+2)^2}-2110 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.7 \begin{align*}{\frac{10}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{130}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1006}{81}\sqrt{1-2\,x}}+{\frac{14}{9\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{149}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{343}{18}\sqrt{1-2\,x}} \right ) }-{\frac{1055\,\sqrt{21}}{243}{\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 = 5.01207, size = 136, normalized size = 1.25 \begin{align*} \frac{10}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{130}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1055}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1006}{81} \, \sqrt{-2 \, x + 1} - \frac{7 \,{\left (149 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64676, size = 266, normalized size = 2.44 \begin{align*} \frac{1055 \, \sqrt{7} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \,{\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt{-2 \, x + 1}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 2.45036, size = 138, normalized size = 1.27 \begin{align*} \frac{10}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{130}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1055}{486} \, \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{1006}{81} \, \sqrt{-2 \, x + 1} - \frac{7 \,{\left (149 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}}{324 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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