Optimal. Leaf size=249 \[ -\frac{23441272 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]
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Rubi [A] time = 0.0971486, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
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)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x} \left (-1200+\frac{1005 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{8 \int \frac{\left (\frac{232425}{4}-\frac{131115 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{16 \int \frac{\frac{7896165}{8}-1154775 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{1964655}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{32 \int \frac{\frac{347150355}{8}-\frac{210824475 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{41257755}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{64 \int \frac{\frac{9262076325}{16}+\frac{7315500075 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{288804285}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{11720636 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1750329}+\frac{780320008 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{19253619}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.300426, size = 115, normalized size = 0.46 \[ \frac{16 \sqrt{2} \left (195080002 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-98384755 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (94808880972 x^5+319217269302 x^4+429993423180 x^3+289719086787 x^2+97637232762 x+13163824553\right )}{(3 x+2)^{11/2}}}{231043428} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{577608570\,{x}^{2}+57760857\,x-173282571} \left ( 47814990930\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-94808880972\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+159383303100\,\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}-316029603240\,\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}+212511070800\,\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}-421372804320\,\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}+141674047200\,\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}-280915202880\,\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}+2844266429160\,{x}^{7}+47224682400\,\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}-93638400960\,\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}+9860944721976\,{x}^{6}+6296624320\,\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 ) -12485120128\,\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 ) +13004174574558\,{x}^{5}+7108597449432\,{x}^{4}-71666565399\,{x}^{3}-1919645346207\,{x}^{2}-839243621199\,x-118474420977 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{11}{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{13}{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}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, 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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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