Optimal. Leaf size=119 \[ \frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (5 x+3)}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0454097, antiderivative size = 119, 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 = {103, 151, 152, 156, 63, 206} \[ \frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (5 x+3)}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx &=\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{1}{7} \int \frac{23-75 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{340}{77 \sqrt{1-2 x} (3+5 x)}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{1}{77} \int \frac{369-3060 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (3+5 x)}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{2 \int \frac{-\frac{56277}{2}+17415 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{5929}\\ &=\frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (3+5 x)}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{1971}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{7875}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (3+5 x)}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{1971}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{7875}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{4644}{5929 \sqrt{1-2 x}}-\frac{340}{77 \sqrt{1-2 x} (3+5 x)}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{1314}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{3150}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0277831, size = 93, normalized size = 0.78 \[ \frac{158994 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7 \left (22050 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+11220 x+7117\right )}{5929 \sqrt{1-2 x} (3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 79, normalized size = 0.7 \begin{align*}{\frac{18}{49}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{1314\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{5929}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{121}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{3150\,\sqrt{55}}{1331}{\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 = 4.02941, size = 161, normalized size = 1.35 \begin{align*} -\frac{1575}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{657}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (17415 \,{\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 77 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68123, size = 421, normalized size = 3.54 \begin{align*} \frac{540225 \, \sqrt{11} \sqrt{5}{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 874467 \, \sqrt{7} \sqrt{3}{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (69660 \, x^{2} + 9696 \, x - 21955\right )} \sqrt{-2 \, x + 1}}{456533 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.50724, size = 178, normalized size = 1.5 \begin{align*} -\frac{1575}{1331} \, \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{657}{343} \, \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 (17415 \,{\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 77 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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