Optimal. Leaf size=127 \[ -\frac{335 \sqrt{1-2 x}}{2 (5 x+3)}+\frac{50 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0463023, antiderivative size = 127, 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} \[ -\frac{335 \sqrt{1-2 x}}{2 (5 x+3)}+\frac{50 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^2} \, dx &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{1}{6} \int \frac{122-167 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{50 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}+\frac{1}{42} \int \frac{9177-10500 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{335 \sqrt{1-2 x}}{2 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{50 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-\frac{1}{462} \int \frac{379071-232155 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{335 \sqrt{1-2 x}}{2 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{50 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}+\frac{6933}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-5610 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{335 \sqrt{1-2 x}}{2 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{50 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-\frac{6933}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+5610 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{335 \sqrt{1-2 x}}{2 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)}+\frac{50 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.103151, size = 90, normalized size = 0.71 \[ -\frac{\sqrt{1-2 x} \left (3015 x^2+3920 x+1271\right )}{2 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 82, normalized size = 0.7 \begin{align*} 18\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{45\, \left ( 1-2\,x \right ) ^{3/2}}{2}}-{\frac{959\,\sqrt{1-2\,x}}{18}} \right ) }-{\frac{2311\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+22\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+204\,{\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 = 1.5427, size = 173, normalized size = 1.36 \begin{align*} -102 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2311}{14} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3015 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 13870 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15939 \, \sqrt{-2 \, x + 1}}{45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73307, size = 392, normalized size = 3.09 \begin{align*} \frac{2311 \, \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 ) + 1428 \, \sqrt{55}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 7 \,{\left (3015 \, x^{2} + 3920 \, x + 1271\right )} \sqrt{-2 \, x + 1}}{14 \,{\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 = 1.51485, size = 166, normalized size = 1.31 \begin{align*} -102 \, \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{2311}{14} \, \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{55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{405 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 959 \, \sqrt{-2 \, x + 1}}{4 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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