Optimal. Leaf size=125 \[ -\frac{12295}{41503 \sqrt{1-2 x}}+\frac{33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac{1115}{1617 (1-2 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.058054, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{12295}{41503 \sqrt{1-2 x}}+\frac{33}{14 (1-2 x)^{3/2} (3 x+2)}-\frac{1115}{1617 (1-2 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^{5/2} (2+3 x)^3 (3+5 x)} \, dx &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{1}{14} \int \frac{7-105 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{1}{98} \int \frac{-1015-5775 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{1115}{1617 (1-2 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac{\int \frac{-\frac{46515}{2}+\frac{351225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{11319}\\ &=-\frac{1115}{1617 (1-2 x)^{3/2}}-\frac{12295}{41503 \sqrt{1-2 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{2 \int \frac{\frac{6679995}{4}-\frac{3872925 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{871563}\\ &=-\frac{1115}{1617 (1-2 x)^{3/2}}-\frac{12295}{41503 \sqrt{1-2 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}-\frac{10935}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{3125}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1115}{1617 (1-2 x)^{3/2}}-\frac{12295}{41503 \sqrt{1-2 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{10935}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{3125}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1115}{1617 (1-2 x)^{3/2}}-\frac{12295}{41503 \sqrt{1-2 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac{3645}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0274896, size = 83, normalized size = 0.66 \[ -\frac{26730 (3 x+2)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7 \left (3500 (3 x+2)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+99 (33 x+23)\right )}{3234 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 84, normalized size = 0.7 \begin{align*} -{\frac{486}{2401\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{27}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{581}{18}\sqrt{1-2\,x}} \right ) }+{\frac{3645\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{11319} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2144}{290521}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{1250\,\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 = 3.2876, size = 173, normalized size = 1.38 \begin{align*} \frac{625}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3645}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{331965 \,{\left (2 \, x - 1\right )}^{3} + 776475 \,{\left (2 \, x - 1\right )}^{2} - 75264 \, x + 46256}{124509 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 49 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32857, size = 489, normalized size = 3.91 \begin{align*} \frac{9003750 \, \sqrt{11} \sqrt{5}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 14554485 \, \sqrt{7} \sqrt{3}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (1327860 \, x^{3} - 438840 \, x^{2} - 594687 \, x + 245383\right )} \sqrt{-2 \, x + 1}}{19174386 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 = 2.38049, size = 173, normalized size = 1.38 \begin{align*} \frac{625}{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{3645}{4802} \, \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{16 \,{\left (804 \, x - 479\right )}}{871563 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{27 \,{\left (243 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 581 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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