Optimal. Leaf size=102 \[ -\frac{68 \sqrt{1-2 x}}{3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0357827, antiderivative size = 102, 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} \[ -\frac{68 \sqrt{1-2 x}}{3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 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)^2} \, dx &=\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}+\frac{1}{3} \int \frac{89-101 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{68 \sqrt{1-2 x}}{3 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-\frac{1}{33} \int \frac{3663-2244 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{68 \sqrt{1-2 x}}{3 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}+469 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-759 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{68 \sqrt{1-2 x}}{3 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-469 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+759 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{68 \sqrt{1-2 x}}{3 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)}-134 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+138 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0513364, size = 105, normalized size = 1.03 \[ \frac{-670 \sqrt{21} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+414 \sqrt{55} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-15 \sqrt{1-2 x} (68 x+43)}{15 (3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 70, normalized size = 0.7 \begin{align*}{\frac{14}{3}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{134\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{22}{5}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{138\,\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 = 2.72442, size = 149, normalized size = 1.46 \begin{align*} -\frac{69}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{67}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (34 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70596, size = 352, normalized size = 3.45 \begin{align*} \frac{207 \, \sqrt{11} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 335 \, \sqrt{7} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 15 \,{\left (68 \, x + 43\right )} \sqrt{-2 \, x + 1}}{15 \,{\left (15 \, x^{2} + 19 \, x + 6\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 = 1.62352, size = 157, normalized size = 1.54 \begin{align*} -\frac{69}{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{67}{3} \, \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{4 \,{\left (34 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 77 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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