Optimal. Leaf size=131 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}+\frac{343 \sqrt{1-2 x}}{9 (3 x+2) (5 x+3)}-\frac{6763 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0474062, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)}+\frac{343 \sqrt{1-2 x}}{9 (3 x+2) (5 x+3)}-\frac{6763 \sqrt{1-2 x}}{18 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^2} \, dx &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{1}{6} \int \frac{(164-97 x) \sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{343 \sqrt{1-2 x}}{9 (2+3 x) (3+5 x)}-\frac{1}{18} \int \frac{-8821+10096 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{6763 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{343 \sqrt{1-2 x}}{9 (2+3 x) (3+5 x)}+\frac{1}{198} \int \frac{-364419+223179 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{6763 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{343 \sqrt{1-2 x}}{9 (2+3 x) (3+5 x)}+\frac{46655}{6} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-12584 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{6763 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{343 \sqrt{1-2 x}}{9 (2+3 x) (3+5 x)}-\frac{46655}{6} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+12584 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{6763 \sqrt{1-2 x}}{18 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)}+\frac{343 \sqrt{1-2 x}}{9 (2+3 x) (3+5 x)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0956148, size = 94, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (20289 x^2+26380 x+8553\right )}{6 (3 x+2)^2 (5 x+3)}-\frac{6665}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+2288 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 82, normalized size = 0.6 \begin{align*} 126\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{131\, \left ( 1-2\,x \right ) ^{3/2}}{18}}-{\frac{931\,\sqrt{1-2\,x}}{54}} \right ) }-{\frac{6665\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{5}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{2288\,\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 = 2.32608, size = 173, normalized size = 1.32 \begin{align*} -\frac{1144}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6665}{18} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20289 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 93338 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 107261 \, \sqrt{-2 \, x + 1}}{3 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35711, size = 421, normalized size = 3.21 \begin{align*} \frac{20592 \, \sqrt{11} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 33325 \, \sqrt{7} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 15 \,{\left (20289 \, x^{2} + 26380 \, x + 8553\right )} \sqrt{-2 \, x + 1}}{90 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 = 2.42602, size = 166, normalized size = 1.27 \begin{align*} -\frac{1144}{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{6665}{18} \, \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{121 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{7 \,{\left (393 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 931 \, \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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