Optimal. Leaf size=186 \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{4819}{440} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac{4270537963 \sqrt{1-2 x} (5 x+3)^{3/2}}{3379200}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac{4270537963 \sqrt{1-2 x} \sqrt{5 x+3}}{409600}+\frac{46975917593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{409600 \sqrt{10}} \]
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Rubi [A] time = 0.0636453, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ \frac{(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{4819}{440} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac{4270537963 \sqrt{1-2 x} (5 x+3)^{3/2}}{3379200}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac{4270537963 \sqrt{1-2 x} \sqrt{5 x+3}}{409600}+\frac{46975917593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{409600 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 153
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac{195 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-11389-\frac{72285 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{(2+3 x) (3+5 x)^{3/2} \left (\frac{7230145}{4}+\frac{22702425 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{1650}\\ &=-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac{4270537963 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{844800}\\ &=-\frac{4270537963 \sqrt{1-2 x} (3+5 x)^{3/2}}{3379200}-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac{4270537963 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{204800}\\ &=-\frac{4270537963 \sqrt{1-2 x} \sqrt{3+5 x}}{409600}-\frac{4270537963 \sqrt{1-2 x} (3+5 x)^{3/2}}{3379200}-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac{46975917593 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{819200}\\ &=-\frac{4270537963 \sqrt{1-2 x} \sqrt{3+5 x}}{409600}-\frac{4270537963 \sqrt{1-2 x} (3+5 x)^{3/2}}{3379200}-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac{46975917593 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{409600 \sqrt{5}}\\ &=-\frac{4270537963 \sqrt{1-2 x} \sqrt{3+5 x}}{409600}-\frac{4270537963 \sqrt{1-2 x} (3+5 x)^{3/2}}{3379200}-\frac{4819}{440} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac{46975917593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{409600 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0913094, size = 89, normalized size = 0.48 \[ \frac{140927752779 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (248832000 x^6+1423526400 x^5+4002203520 x^4+8217694800 x^3+18987469764 x^2-58600061024 x+21368105901\right )}{12288000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 188, normalized size = 1. \begin{align*}{\frac{1}{24576000\, \left ( 2\,x-1 \right ) ^{2}} \left ( -4976640000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-28470528000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-80044070400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+563711011116\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-164353896000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-563711011116\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-379749395280\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+140927752779\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1172001220480\,x\sqrt{-10\,{x}^{2}-x+3}-427362118020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.01534, size = 478, normalized size = 2.57 \begin{align*} -\frac{81}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{891}{256} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11872553}{2048} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{514294407}{8192000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{139491}{5120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{189 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (2 \, x - 1\right )}} - \frac{4250367}{20480} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{89257707}{409600} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{800415}{512} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{132055 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{384 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{56595 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{64 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{24255 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (2 \, x - 1\right )}} + \frac{1452605 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{15827735 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55, size = 406, normalized size = 2.18 \begin{align*} -\frac{140927752779 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (248832000 \, x^{6} + 1423526400 \, x^{5} + 4002203520 \, x^{4} + 8217694800 \, x^{3} + 18987469764 \, x^{2} - 58600061024 \, x + 21368105901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{24576000 \,{\left (4 \, x^{2} - 4 \, x + 1\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 = 2.30169, size = 166, normalized size = 0.89 \begin{align*} \frac{46975917593}{4096000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \,{\left (72 \,{\left (4 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 509 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 20743 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 18487133 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4270537963 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 469759175930 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7751026402845 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{768000000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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