Optimal. Leaf size=222 \[ -\frac{1241596 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{750141}+\frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]
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Rubi [A] time = 0.0778519, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]
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)^{5/2}}{(2+3 x)^{11/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{4}{567} \int \frac{\left (-1120-\frac{1625 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{2776 \sqrt{1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac{8 \int \frac{(3+5 x)^{3/2} \left (\frac{28945}{4}+\frac{9225 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx}{8505}\\ &=-\frac{13316 \sqrt{1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{2776 \sqrt{1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac{16 \int \frac{\sqrt{3+5 x} \left (\frac{2510595}{8}+\frac{359475 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx}{535815}\\ &=-\frac{1241596 \sqrt{1-2 x} \sqrt{3+5 x}}{750141 \sqrt{2+3 x}}-\frac{13316 \sqrt{1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{2776 \sqrt{1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac{32 \int \frac{\frac{53475825}{16}+\frac{1883325 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{11252115}\\ &=-\frac{1241596 \sqrt{1-2 x} \sqrt{3+5 x}}{750141 \sqrt{2+3 x}}-\frac{13316 \sqrt{1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{2776 \sqrt{1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}+\frac{100444 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{750141}+\frac{6828778 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{750141}\\ &=-\frac{1241596 \sqrt{1-2 x} \sqrt{3+5 x}}{750141 \sqrt{2+3 x}}-\frac{13316 \sqrt{1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{2776 \sqrt{1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}\\ \end{align*}
Mathematica [A] time = 0.178896, size = 109, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (10192945 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+50222 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (29072682 x^4+115002639 x^3+142557831 x^2+71920155 x+12903031\right )}{(3 x+2)^{9/2}}\right )}{2250423} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 504, normalized size = 2.3 \begin{align*} -{\frac{2}{22504230\,{x}^{2}+2250423\,x-6751269} \left ( 825628545\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+4067982\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+2201676120\,\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}+10847952\,\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}+2201676120\,\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}+10847952\,\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}+978522720\,\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}+4821312\,\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}-872180460\,{x}^{6}+163087120\,\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 ) +803552\,\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 ) -3537297216\,{x}^{5}-4360088709\,{x}^{4}-1550254392\,{x}^{3}+680169084\,{x}^{2}+608572302\,x+116127279 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{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{11}{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}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, 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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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