Optimal. Leaf size=118 \[ \frac{207 \sqrt{1-2 x}}{2 (5 x+3)}-\frac{103 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
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Rubi [A] time = 0.0434747, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{207 \sqrt{1-2 x}}{2 (5 x+3)}-\frac{103 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
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)^2 (3+5 x)^3} \, dx &=\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{1}{3} \int \frac{124-171 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{103 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}-\frac{1}{66} \int \frac{8910-10197 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{103 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{207 \sqrt{1-2 x}}{2 (3+5 x)}+\frac{1}{726} \int \frac{368082-225423 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{103 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{207 \sqrt{1-2 x}}{2 (3+5 x)}-2142 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{6933}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{103 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{207 \sqrt{1-2 x}}{2 (3+5 x)}+2142 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{6933}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{103 \sqrt{1-2 x}}{6 (3+5 x)^2}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{207 \sqrt{1-2 x}}{2 (3+5 x)}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0766335, size = 88, normalized size = 0.75 \[ \frac{\sqrt{1-2 x} \left (3105 x^2+3830 x+1178\right )}{2 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 82, normalized size = 0.7 \begin{align*} -14\,{\frac{\sqrt{1-2\,x}}{-2\,x-4/3}}+204\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+50\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{137\, \left ( 1-2\,x \right ) ^{3/2}}{10}}+{\frac{297\,\sqrt{1-2\,x}}{10}} \right ) }-{\frac{6933\,\sqrt{55}}{55}{\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 = 2.72005, size = 171, normalized size = 1.45 \begin{align*} \frac{6933}{110} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - 102 \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{3105 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 13870 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15477 \, \sqrt{-2 \, x + 1}}{75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63368, size = 381, normalized size = 3.23 \begin{align*} \frac{6933 \, \sqrt{55}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11220 \, \sqrt{21}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 55 \,{\left (3105 \, x^{2} + 3830 \, x + 1178\right )} \sqrt{-2 \, x + 1}}{110 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 = 2.04946, size = 166, normalized size = 1.41 \begin{align*} \frac{6933}{110} \, \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 ) - 102 \, \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{21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{5 \,{\left (137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 297 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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