Optimal. Leaf size=222 \[ \frac{270668 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{35721}-\frac{1844 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{62596 \sqrt{1-2 x} (5 x+3)^{3/2}}{3969 \sqrt{3 x+2}}+\frac{1353340 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35721}-\frac{904798 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721} \]
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Rubi [A] time = 0.0845279, 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} \[ -\frac{1844 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{62596 \sqrt{1-2 x} (5 x+3)^{3/2}}{3969 \sqrt{3 x+2}}+\frac{1353340 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{35721}+\frac{270668 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721}-\frac{904798 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721} \]
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{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{\left (-\frac{1415}{2}-\frac{3275 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{1844 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac{8 \int \frac{\left (\frac{19045}{4}-22200 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx}{2835}\\ &=-\frac{62596 \sqrt{1-2 x} (3+5 x)^{3/2}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{1844 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac{16 \int \frac{\left (\frac{1656225}{8}-\frac{5075025 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{59535}\\ &=\frac{1353340 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35721}-\frac{62596 \sqrt{1-2 x} (3+5 x)^{3/2}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{1844 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac{16 \int \frac{-\frac{2298225}{2}-\frac{33929925 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{535815}\\ &=\frac{1353340 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35721}-\frac{62596 \sqrt{1-2 x} (3+5 x)^{3/2}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{1844 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac{904798 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{35721}-\frac{1488674 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{35721}\\ &=\frac{1353340 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{35721}-\frac{62596 \sqrt{1-2 x} (3+5 x)^{3/2}}{3969 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac{1844 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac{904798 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721}+\frac{270668 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{35721}\\ \end{align*}
Mathematica [A] time = 0.170721, size = 109, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (452399 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2685410 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (396900 x^4+9846603 x^3+17788023 x^2+11107911 x+2337569\right )}{(3 x+2)^{7/2}}\right )}{107163} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 414, normalized size = 1.9 \begin{align*}{\frac{2}{1071630\,{x}^{2}+107163\,x-321489} \left ( 72506070\,\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}-12214773\,\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}+145012140\,\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}-24429546\,\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}+96674760\,\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}-16286364\,\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}+11907000\,{x}^{6}+21483280\,\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 ) -3619192\,\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 ) +296588790\,{x}^{5}+559608399\,{x}^{4}+297981972\,{x}^{3}-56641404\,{x}^{2}-92958492\,x-21038121 \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{5}{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 (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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{5}{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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