Optimal. Leaf size=135 \[ \frac{55 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{220}{21} (1-2 x)^{3/2} (5 x+3)^2+\frac{55 (1-2 x)^{3/2} (603 x+209)}{1134}-\frac{935}{81} \sqrt{1-2 x}+\frac{935}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0617776, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {97, 12, 149, 153, 147, 50, 63, 206} \[ \frac{55 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{220}{21} (1-2 x)^{3/2} (5 x+3)^2+\frac{55 (1-2 x)^{3/2} (603 x+209)}{1134}-\frac{935}{81} \sqrt{1-2 x}+\frac{935}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 12
Rule 149
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{1}{6} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}-\frac{55}{6} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac{55}{18} \int \frac{\sqrt{1-2 x} (3+5 x)^2 (-3+72 x)}{2+3 x} \, dx\\ &=-\frac{220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}-\frac{55}{378} \int \frac{\sqrt{1-2 x} (3+5 x) (45+603 x)}{2+3 x} \, dx\\ &=-\frac{220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac{55 (1-2 x)^{3/2} (209+603 x)}{1134}-\frac{935}{54} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{935}{81} \sqrt{1-2 x}-\frac{220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac{55 (1-2 x)^{3/2} (209+603 x)}{1134}-\frac{6545}{162} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{935}{81} \sqrt{1-2 x}-\frac{220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac{55 (1-2 x)^{3/2} (209+603 x)}{1134}+\frac{6545}{162} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{935}{81} \sqrt{1-2 x}-\frac{220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac{55 (1-2 x)^{3/2} (209+603 x)}{1134}+\frac{935}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0434734, size = 75, normalized size = 0.56 \[ \frac{\sqrt{1-2 x} \left (54000 x^5-24120 x^4-17460 x^3-67962 x^2-152833 x-64943\right )}{1134 (3 x+2)^2}+\frac{935}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, normalized size = 0.6 \begin{align*} -{\frac{125}{189} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{10}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{370}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{8198}{729}\sqrt{1-2\,x}}-{\frac{14}{81\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{73}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1519}{18}\sqrt{1-2\,x}} \right ) }+{\frac{935\,\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 = 4.98348, size = 149, normalized size = 1.1 \begin{align*} -\frac{125}{189} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{10}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{370}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{935}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8198}{729} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (657 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1519 \, \sqrt{-2 \, x + 1}\right )}}{729 \,{\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 = 1.57745, size = 289, normalized size = 2.14 \begin{align*} \frac{6545 \, \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 (54000 \, x^{5} - 24120 \, x^{4} - 17460 \, x^{3} - 67962 \, x^{2} - 152833 \, x - 64943\right )} \sqrt{-2 \, x + 1}}{3402 \,{\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 = 2.11914, size = 159, normalized size = 1.18 \begin{align*} \frac{125}{189} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{10}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{370}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{935}{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{8198}{729} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (657 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1519 \, \sqrt{-2 \, x + 1}\right )}}{2916 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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