Optimal. Leaf size=128 \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
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Rubi [A] time = 0.0436493, antiderivative size = 128, 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 = {97, 153, 147, 50, 63, 206} \[ -\frac{(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac{11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac{2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac{(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac{258 \sqrt{1-2 x}}{15625}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac{1}{5} \int \frac{(6-33 x) \sqrt{1-2 x} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac{1}{225} \int \frac{(-243-18 x) \sqrt{1-2 x} (2+3 x)^2}{3+5 x} \, dx\\ &=-\frac{2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac{\int \frac{\sqrt{1-2 x} (2+3 x) (17010+25641 x)}{3+5 x} \, dx}{7875}\\ &=-\frac{2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac{129 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac{258 \sqrt{1-2 x}}{15625}-\frac{2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac{1419 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac{258 \sqrt{1-2 x}}{15625}-\frac{2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac{1419 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{15625}\\ &=\frac{258 \sqrt{1-2 x}}{15625}-\frac{2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac{11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac{(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac{258 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625}\\ \end{align*}
Mathematica [A] time = 0.0771411, size = 78, normalized size = 0.61 \[ -\frac{5 \sqrt{1-2 x} \left (787500 x^5+1395000 x^4+157275 x^3-924335 x^2-143235 x+161312\right )+1806 \sqrt{55} (5 x+3) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{546875 (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 81, normalized size = 0.6 \begin{align*} -{\frac{9}{100} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{999}{1750} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{12393}{12500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{8}{3125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{52}{3125}\sqrt{1-2\,x}}+{\frac{22}{78125}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{258\,\sqrt{55}}{78125}{\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 = 3.5273, size = 132, normalized size = 1.03 \begin{align*} -\frac{9}{100} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{999}{1750} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{12393}{12500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60319, size = 275, normalized size = 2.15 \begin{align*} \frac{903 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \,{\left (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt{-2 \, x + 1}}{546875 \,{\left (5 \, x + 3\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 = 2.01928, size = 165, normalized size = 1.29 \begin{align*} -\frac{9}{100} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{999}{1750} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{12393}{12500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{8}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{129}{78125} \, \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{52}{3125} \, \sqrt{-2 \, x + 1} - \frac{11 \, \sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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