Optimal. Leaf size=222 \[ -\frac{722133 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3500}+\frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}-\frac{1341}{154} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{140289 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{3850}-\frac{2166399 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{7700}-\frac{6547351 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]
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Rubi [A] time = 0.0833216, antiderivative size = 222, 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} \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}-\frac{1341}{154} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{140289 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{3850}-\frac{2166399 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{7700}-\frac{722133 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}-\frac{6547351 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]
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)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^{5/2} \sqrt{3+5 x} \left (\frac{93}{2}+75 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-6285-\frac{20115 x}{2}\right ) (2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{1341}{154} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{2664975}{4}+\frac{2104335 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{1155}\\ &=-\frac{140289 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3850}-\frac{1341}{154} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{\int \frac{\left (-\frac{190065795}{4}-\frac{292463865 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{28875}\\ &=-\frac{2166399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7700}-\frac{140289 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3850}-\frac{1341}{154} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\frac{12310799985}{8}+\frac{9722816235 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{259875}\\ &=-\frac{2166399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7700}-\frac{140289 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3850}-\frac{1341}{154} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{2166399 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{7000}+\frac{6547351 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3500}\\ &=-\frac{2166399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7700}-\frac{140289 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{3850}-\frac{1341}{154} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{6547351 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}-\frac{722133 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}\\ \end{align*}
Mathematica [A] time = 0.254948, size = 130, normalized size = 0.59 \[ -\frac{-6595505 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (40500 x^4+198180 x^3+567906 x^2-2751916 x+1041609\right )+13094702 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{21000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 243, normalized size = 1.1 \begin{align*}{\frac{1}{21000\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 13191010\,\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}-26189404\,\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}-6075000\,{x}^{6}-6595505\,\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 ) +13094702\,\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 ) -37422000\,{x}^{5}-125270100\,{x}^{4}+292994460\,{x}^{3}+332548330\,{x}^{2}-32790750\,x-62496540 \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{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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 (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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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