Optimal. Leaf size=249 \[ -\frac{363103712 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1452605 \sqrt{33}}+\frac{12071114168 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}-\frac{181551856 \sqrt{1-2 x} \sqrt{3 x+2}}{871563 (5 x+3)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{12071114168 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}} \]
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Rubi [A] time = 0.0999338, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{12071114168 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}-\frac{181551856 \sqrt{1-2 x} \sqrt{3 x+2}}{871563 (5 x+3)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{363103712 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}}-\frac{12071114168 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{2}{77} \int \frac{-\frac{203}{2}-135 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac{4 \int \frac{-\frac{3277}{2}+\frac{2415 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx}{2695}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{8 \int \frac{-\frac{540213}{4}+\frac{366525 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{56595}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16 \int \frac{-\frac{40301295}{4}+\frac{46301085 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{396165}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{181551856 \sqrt{1-2 x} \sqrt{2+3 x}}{871563 (3+5 x)^{3/2}}+\frac{32 \int \frac{-\frac{3301192785}{8}+\frac{510614595 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{13073445}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{181551856 \sqrt{1-2 x} \sqrt{2+3 x}}{871563 (3+5 x)^{3/2}}+\frac{12071114168 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}-\frac{64 \int \frac{-\frac{42986714535}{8}-\frac{67900017195 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{143807895}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{181551856 \sqrt{1-2 x} \sqrt{2+3 x}}{871563 (3+5 x)^{3/2}}+\frac{12071114168 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}+\frac{181551856 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1452605}+\frac{12071114168 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{15978655}\\ &=\frac{4}{77 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{181551856 \sqrt{1-2 x} \sqrt{2+3 x}}{871563 (3+5 x)^{3/2}}+\frac{12071114168 \sqrt{1-2 x} \sqrt{2+3 x}}{9587193 \sqrt{3+5 x}}-\frac{12071114168 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}}-\frac{363103712 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.282295, size = 114, normalized size = 0.46 \[ \frac{2 \left (4 \sqrt{2} \left (1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-759987865 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{-8148002063400 x^5-16841199826980 x^4-9658241620704 x^3+1466692421066 x^2+2920885694212 x+687365548973}{\sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{47935965} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 406, normalized size = 1.6 \begin{align*} -{\frac{2}{95871930\,x-47935965}\sqrt{1-2\,x} \left ( 271600068780\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-136797815700\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+525093466308\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-264475777020\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+337991196704\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-170237281760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+72426685008\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -36479417520\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -8148002063400\,{x}^{5}-16841199826980\,{x}^{4}-9658241620704\,{x}^{3}+1466692421066\,{x}^{2}+2920885694212\,x+687365548973 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{40500 \, x^{9} + 140400 \, x^{8} + 175365 \, x^{7} + 66873 \, x^{6} - 46885 \, x^{5} - 52853 \, x^{4} - 11968 \, x^{3} + 5112 \, x^{2} + 3024 \, x + 432}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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