Optimal. Leaf size=93 \[ \frac{65 \sqrt{1-2 x}}{6 (3 x+2)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0336983, antiderivative size = 93, 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, 151, 156, 63, 206} \[ \frac{65 \sqrt{1-2 x}}{6 (3 x+2)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)} \, dx &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{87-97 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2}+\frac{65 \sqrt{1-2 x}}{6 (2+3 x)}+\frac{1}{42} \int \frac{3717-2275 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2}+\frac{65 \sqrt{1-2 x}}{6 (2+3 x)}-\frac{2243}{6} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+605 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2}+\frac{65 \sqrt{1-2 x}}{6 (2+3 x)}+\frac{2243}{6} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-605 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2}+\frac{65 \sqrt{1-2 x}}{6 (2+3 x)}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0708349, size = 78, normalized size = 0.84 \[ \frac{\sqrt{1-2 x} (195 x+137)}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -18\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{65\, \left ( 1-2\,x \right ) ^{3/2}}{18}}-{\frac{469\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{2243\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-22\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52893, size = 149, normalized size = 1.6 \begin{align*} 11 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2243}{126} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 469 \, \sqrt{-2 \, x + 1}}{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 = 1.43565, size = 313, normalized size = 3.37 \begin{align*} \frac{1386 \, \sqrt{55}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 2243 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (195 \, x + 137\right )} \sqrt{-2 \, x + 1}}{126 \,{\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 = 1.77544, size = 144, normalized size = 1.55 \begin{align*} 11 \, \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{2243}{126} \, \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{195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 469 \, \sqrt{-2 \, x + 1}}{12 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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