Optimal. Leaf size=191 \[ -\frac{4244 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3969}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{63 (3 x+2)^{5/2}}+\frac{608 (5 x+3)^{3/2} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{4244 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 \sqrt{3 x+2}}-\frac{11576 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969} \]
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Rubi [A] time = 0.0662442, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 150, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac{46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{63 (3 x+2)^{5/2}}+\frac{608 (5 x+3)^{3/2} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}-\frac{4244 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 \sqrt{3 x+2}}-\frac{4244 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969}-\frac{11576 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{63 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{\left (-705-\frac{975 x}{2}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{63 (2+3 x)^{5/2}}+\frac{608 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{3/2}}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{16605}{4}+\frac{8475 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx}{2835}\\ &=-\frac{4244 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{63 (2+3 x)^{5/2}}+\frac{608 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{3/2}}+\frac{16 \int \frac{\frac{435525}{8}+\frac{108525 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{59535}\\ &=-\frac{4244 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{63 (2+3 x)^{5/2}}+\frac{608 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{3/2}}+\frac{11576 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3969}+\frac{23342 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3969}\\ &=-\frac{4244 \sqrt{1-2 x} \sqrt{3+5 x}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac{46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{63 (2+3 x)^{5/2}}+\frac{608 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{3/2}}-\frac{11576 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969}-\frac{4244 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3969}\\ \end{align*}
Mathematica [A] time = 0.159218, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (29225 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+5788 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (182736 x^3+409005 x^2+292578 x+67759\right )}{(3 x+2)^{7/2}}\right )}{11907} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 409, normalized size = 2.1 \begin{align*} -{\frac{2}{119070\,{x}^{2}+11907\,x-35721} \left ( 789075\,\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}+156276\,\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}+1578150\,\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}+312552\,\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}+1052100\,\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}+208368\,\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}+233800\,\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 ) +46304\,\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 ) -5482080\,{x}^{5}-12818358\,{x}^{4}-8359731\,{x}^{3}+770541\,{x}^{2}+2429925\,x+609831 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{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 (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, 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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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