Optimal. Leaf size=108 \[ -\frac{(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac{5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac{10}{63} (1-2 x)^{3/2} (27 x+22)+\frac{8}{9} \sqrt{1-2 x}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0320163, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 153, 147, 50, 63, 206} \[ -\frac{(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac{5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac{10}{63} (1-2 x)^{3/2} (27 x+22)+\frac{8}{9} \sqrt{1-2 x}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}+\frac{1}{3} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{2+3 x} \, dx\\ &=\frac{5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac{1}{63} \int \frac{(-288-810 x) \sqrt{1-2 x} (3+5 x)}{2+3 x} \, dx\\ &=\frac{5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac{10}{63} (1-2 x)^{3/2} (22+27 x)+\frac{4}{3} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{8}{9} \sqrt{1-2 x}+\frac{5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac{10}{63} (1-2 x)^{3/2} (22+27 x)+\frac{28}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{8}{9} \sqrt{1-2 x}+\frac{5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac{10}{63} (1-2 x)^{3/2} (22+27 x)-\frac{28}{9} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{8}{9} \sqrt{1-2 x}+\frac{5}{7} (1-2 x)^{3/2} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{3 (2+3 x)}-\frac{10}{63} (1-2 x)^{3/2} (22+27 x)-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0433867, size = 70, normalized size = 0.65 \[ -\frac{\sqrt{1-2 x} \left (1500 x^4+780 x^3-1005 x^2-442 x+85\right )}{63 (3 x+2)}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 72, normalized size = 0.7 \begin{align*}{\frac{125}{126} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{145}{54} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{10}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{214}{243}\sqrt{1-2\,x}}-{\frac{14}{729}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{8\,\sqrt{21}}{27}{\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 = 1.77511, size = 120, normalized size = 1.11 \begin{align*} \frac{125}{126} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{145}{54} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{10}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{214}{243} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2913, size = 230, normalized size = 2.13 \begin{align*} \frac{28 \, \sqrt{7} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 3 \,{\left (1500 \, x^{4} + 780 \, x^{3} - 1005 \, x^{2} - 442 \, x + 85\right )} \sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\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 = 2.30716, size = 143, normalized size = 1.32 \begin{align*} -\frac{125}{126} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{145}{54} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{10}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{27} \, \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{214}{243} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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