Optimal. Leaf size=147 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac{2721 \sqrt{1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac{377748 \sqrt{1-2 x} (3 x+2)^2}{831875}+\frac{63 \sqrt{1-2 x} (831375 x+2492512)}{8318750}-\frac{33873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375 \sqrt{55}} \]
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Rubi [A] time = 0.0512524, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 153, 147, 63, 206} \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac{2721 \sqrt{1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac{377748 \sqrt{1-2 x} (3 x+2)^2}{831875}+\frac{63 \sqrt{1-2 x} (831375 x+2492512)}{8318750}-\frac{33873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1}{11} \int \frac{(2+3 x)^4 (173+312 x)}{\sqrt{1-2 x} (3+5 x)^3} \, dx\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{\int \frac{(2+3 x)^3 (12450+21657 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{2721 \sqrt{1-2 x} (2+3 x)^3}{66550 (3+5 x)}-\frac{\int \frac{(2+3 x)^2 (446523+755496 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{66550}\\ &=\frac{377748 \sqrt{1-2 x} (2+3 x)^2}{831875}-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{2721 \sqrt{1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac{\int \frac{(-31392102-52376625 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{1663750}\\ &=\frac{377748 \sqrt{1-2 x} (2+3 x)^2}{831875}-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{2721 \sqrt{1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac{63 \sqrt{1-2 x} (2492512+831375 x)}{8318750}+\frac{33873 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{8318750}\\ &=\frac{377748 \sqrt{1-2 x} (2+3 x)^2}{831875}-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{2721 \sqrt{1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac{63 \sqrt{1-2 x} (2492512+831375 x)}{8318750}-\frac{33873 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{8318750}\\ &=\frac{377748 \sqrt{1-2 x} (2+3 x)^2}{831875}-\frac{71 \sqrt{1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^5}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{2721 \sqrt{1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac{63 \sqrt{1-2 x} (2492512+831375 x)}{8318750}-\frac{33873 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4159375 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.107978, size = 101, normalized size = 0.69 \[ \frac{\frac{13230 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{\sqrt{1-2 x}}-\frac{55 \left (22052250 x^5+129373200 x^4+516610710 x^3+69300960 x^2-377289427 x-154786070\right )}{\sqrt{1-2 x} (5 x+3)^2}-4956 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{41593750} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, normalized size = 0.6 \begin{align*}{\frac{729}{5000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{8991}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{333639}{25000}\sqrt{1-2\,x}}+{\frac{117649}{10648}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{166375\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{403}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{891}{10}\sqrt{1-2\,x}} \right ) }-{\frac{33873\,\sqrt{55}}{228765625}{\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 = 2.06202, size = 149, normalized size = 1.01 \begin{align*} \frac{729}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{8991}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33873}{457531250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{333639}{25000} \, \sqrt{-2 \, x + 1} + \frac{1838268849 \,{\left (2 \, x - 1\right )}^{2} + 16176751756 \, x + 808829747}{6655000 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62173, size = 342, normalized size = 2.33 \begin{align*} \frac{33873 \, \sqrt{55}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (242574750 \, x^{5} + 1423105200 \, x^{4} + 5682717810 \, x^{3} + 762244410 \, x^{2} - 4150263077 \, x - 1702670584\right )} \sqrt{-2 \, x + 1}}{457531250 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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 = 2.47843, size = 150, normalized size = 1.02 \begin{align*} \frac{729}{5000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{8991}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{33873}{457531250} \, \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{333639}{25000} \, \sqrt{-2 \, x + 1} + \frac{117649}{10648 \, \sqrt{-2 \, x + 1}} + \frac{403 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 891 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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