Optimal. Leaf size=158 \[ \frac{106 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{19 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{5 x+3}}{147 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{19}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0491291, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac{19 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{5 x+3}}{147 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{106 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{19}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{1}{21} \int \frac{\frac{29}{2}+15 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{8 \sqrt{3+5 x}}{147 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{2 \int \frac{-\frac{1089}{4}-330 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1617}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{8 \sqrt{3+5 x}}{147 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{19 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{4 \int \frac{-\frac{1815}{2}-\frac{3135 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11319}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{8 \sqrt{3+5 x}}{147 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{19 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}-\frac{19}{343} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx-\frac{53}{343} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} \sqrt{2+3 x}}-\frac{8 \sqrt{3+5 x}}{147 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{19 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{19}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{106 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.143724, size = 100, normalized size = 0.63 \[ \frac{-\sqrt{2} \left (140 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+19 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{2 \sqrt{5 x+3} \left (114 x^2-170 x-213\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}}{1029} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 228, normalized size = 1.4 \begin{align*}{\frac{1}{1029\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 280\,\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}+38\,\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}-140\,\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 ) -19\,\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 ) -1140\,{x}^{3}+1016\,{x}^{2}+3150\,x+1278 \right ) } \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{3}{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 (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{72 \, x^{5} - 12 \, x^{4} - 58 \, x^{3} + 15 \, x^{2} + 12 \, x - 4}, 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{3}{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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