Optimal. Leaf size=160 \[ \frac{536}{45} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{17804 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{17804}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0512834, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt{5 x+3}}-\frac{17804 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{536}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{17804}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{2}{9} \int \frac{(130-29 x) \sqrt{1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{4}{27} \int \frac{-\frac{7139}{2}+2211 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{17804 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}+\frac{8}{297} \int \frac{-46497-\frac{146883 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{17804 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}-\frac{2948}{45} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{17804}{45} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{1792 \sqrt{1-2 x}}{27 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{17804 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}+\frac{17804}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{536}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.128376, size = 100, normalized size = 0.62 \[ -\frac{4}{135} \sqrt{2} \left (4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2240 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{2 \sqrt{1-2 x} \left (26706 x^2+34726 x+11265\right )}{9 (3 x+2)^{3/2} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 219, normalized size = 1.4 \begin{align*} -{\frac{2}{1350\,{x}^{2}+135\,x-405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13440\,\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}-26706\,\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}+8960\,\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 ) -17804\,\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 ) +801180\,{x}^{3}+641190\,{x}^{2}-182940\,x-168975 \right ) \left ( 2+3\,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{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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