Optimal. Leaf size=95 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{49 \sqrt{1-2 x}}{2 (3 x+2)}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0364551, antiderivative size = 95, 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 = {98, 149, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{49 \sqrt{1-2 x}}{2 (3 x+2)}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)} \, dx &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{(129-27 x) \sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac{49 \sqrt{1-2 x}}{2 (2+3 x)}-\frac{1}{18} \int \frac{-3501+2151 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac{49 \sqrt{1-2 x}}{2 (2+3 x)}-\frac{1645}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+1331 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac{49 \sqrt{1-2 x}}{2 (2+3 x)}+\frac{1645}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-1331 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2}+\frac{49 \sqrt{1-2 x}}{2 (2+3 x)}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0669108, size = 80, normalized size = 0.84 \[ \frac{7 \sqrt{1-2 x} (61 x+43)}{6 (3 x+2)^2}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -126\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{61\, \left ( 1-2\,x \right ) ^{3/2}}{54}}-{\frac{49\,\sqrt{1-2\,x}}{18}} \right ) }+{\frac{235\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{242\,\sqrt{55}}{5}{\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 = 2.02752, size = 149, normalized size = 1.57 \begin{align*} \frac{121}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{235}{6} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7 \,{\left (61 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 147 \, \sqrt{-2 \, x + 1}\right )}}{3 \,{\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 = 1.34998, size = 350, normalized size = 3.68 \begin{align*} \frac{726 \, \sqrt{11} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1175 \, \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 ) + 35 \,{\left (61 \, x + 43\right )} \sqrt{-2 \, x + 1}}{30 \,{\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 = 2.17757, size = 144, normalized size = 1.52 \begin{align*} \frac{121}{5} \, \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{235}{6} \, \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{7 \,{\left (61 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 147 \, \sqrt{-2 \, x + 1}\right )}}{12 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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