Optimal. Leaf size=140 \[ -\frac{129 \sqrt{1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{2643 \sqrt{1-2 x} (3 x+2)^3}{1750}+\frac{1404 \sqrt{1-2 x} (3 x+2)^2}{3125}+\frac{9 \sqrt{1-2 x} (1375 x+32)}{31250}-\frac{12279 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]
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Rubi [A] time = 0.0485552, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 153, 147, 63, 206} \[ -\frac{129 \sqrt{1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac{2643 \sqrt{1-2 x} (3 x+2)^3}{1750}+\frac{1404 \sqrt{1-2 x} (3 x+2)^2}{3125}+\frac{9 \sqrt{1-2 x} (1375 x+32)}{31250}-\frac{12279 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx &=-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}+\frac{1}{10} \int \frac{(6-33 x) \sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac{1}{50} \int \frac{(870-2643 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{2643 \sqrt{1-2 x} (2+3 x)^3}{1750}-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}-\frac{\int \frac{(2+3 x)^2 (-5397+19656 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{1750}\\ &=\frac{1404 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{2643 \sqrt{1-2 x} (2+3 x)^3}{1750}-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac{\int \frac{(33978-86625 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{43750}\\ &=\frac{1404 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{2643 \sqrt{1-2 x} (2+3 x)^3}{1750}-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac{9 \sqrt{1-2 x} (32+1375 x)}{31250}+\frac{12279 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{31250}\\ &=\frac{1404 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{2643 \sqrt{1-2 x} (2+3 x)^3}{1750}-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac{9 \sqrt{1-2 x} (32+1375 x)}{31250}-\frac{12279 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{31250}\\ &=\frac{1404 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{2643 \sqrt{1-2 x} (2+3 x)^3}{1750}-\frac{(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac{129 \sqrt{1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac{9 \sqrt{1-2 x} (32+1375 x)}{31250}-\frac{12279 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0789423, size = 73, normalized size = 0.52 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (2025000 x^5+3267000 x^4-496350 x^3-2120880 x^2-489445 x+96776\right )}{(5 x+3)^2}-171906 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12031250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 84, normalized size = 0.6 \begin{align*}{\frac{81}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{1107}{6250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{36}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{228}{3125}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{259}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2871}{500}\sqrt{1-2\,x}} \right ) }-{\frac{12279\,\sqrt{55}}{859375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.10782, size = 149, normalized size = 1.06 \begin{align*} \frac{81}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1107}{6250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{36}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12279}{1718750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{228}{3125} \, \sqrt{-2 \, x + 1} + \frac{1295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2871 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59753, size = 289, normalized size = 2.06 \begin{align*} \frac{85953 \, \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 (2025000 \, x^{5} + 3267000 \, x^{4} - 496350 \, x^{3} - 2120880 \, x^{2} - 489445 \, x + 96776\right )} \sqrt{-2 \, x + 1}}{12031250 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 1.44398, size = 159, normalized size = 1.14 \begin{align*} -\frac{81}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1107}{6250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{36}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12279}{1718750} \, \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{228}{3125} \, \sqrt{-2 \, x + 1} + \frac{1295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2871 \, \sqrt{-2 \, x + 1}}{62500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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