Optimal. Leaf size=218 \[ -\frac{6770629 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{17500 \sqrt{33}}+\frac{\sqrt{5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac{166 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 \sqrt{1-2 x}}-\frac{1327}{154} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{139163 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3850}-\frac{6478333 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38500}-\frac{112543103 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}} \]
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Rubi [A] time = 0.0803666, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac{\sqrt{5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac{166 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 \sqrt{1-2 x}}-\frac{1327}{154} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{139163 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3850}-\frac{6478333 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{38500}-\frac{6770629 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500 \sqrt{33}}-\frac{112543103 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^{7/2} \left (\frac{91}{2}+75 x\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-6054-\frac{19905 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1327}{154} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{(2+3 x)^{3/2} \left (\frac{2551035}{4}+\frac{2087445 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1155}\\ &=-\frac{139163 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3850}-\frac{1327}{154} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{\int \frac{\left (-\frac{179737875}{4}-\frac{291524985 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{28875}\\ &=-\frac{6478333 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{38500}-\frac{139163 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3850}-\frac{1327}{154} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\frac{12824947395}{8}+\frac{5064439635 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{433125}\\ &=-\frac{6478333 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{38500}-\frac{139163 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3850}-\frac{1327}{154} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}+\frac{6770629 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{35000}+\frac{112543103 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{96250}\\ &=-\frac{6478333 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{38500}-\frac{139163 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3850}-\frac{1327}{154} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{166 (2+3 x)^{7/2} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{9/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{112543103 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}}-\frac{6770629 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.251695, size = 130, normalized size = 0.6 \[ -\frac{-226741655 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (1336500 x^4+6664680 x^3+19375686 x^2-94671446 x+35797779\right )+450172412 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1155000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 243, normalized size = 1.1 \begin{align*}{\frac{1}{1155000\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 453483310\,\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}-900344824\,\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}-200475000\,{x}^{6}-226741655\,\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 ) +450172412\,\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 ) -1253637000\,{x}^{5}-4252832100\,{x}^{4}+10119455760\,{x}^{3}+11455366730\,{x}^{2}-1121291250\,x-2147866740 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{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{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, 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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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