Optimal. Leaf size=222 \[ -\frac{1717916 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8751645}+\frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{1864 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{5 x+3}}{8751645 \sqrt{3 x+2}}-\frac{558524 \sqrt{1-2 x} \sqrt{5 x+3}}{1250235 (3 x+2)^{3/2}}-\frac{17830424 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645} \]
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Rubi [A] time = 0.0799221, 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, 152, 158, 113, 119} \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{1864 \sqrt{1-2 x} (5 x+3)^{3/2}}{6615 (3 x+2)^{5/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{5 x+3}}{8751645 \sqrt{3 x+2}}-\frac{558524 \sqrt{1-2 x} \sqrt{5 x+3}}{1250235 (3 x+2)^{3/2}}-\frac{1717916 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645}-\frac{17830424 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645} \]
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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{2}{27} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{4}{567} \int \frac{(3+5 x)^{3/2} \left (-584+\frac{345 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac{1864 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{8 \int \frac{\sqrt{3+5 x} \left (-\frac{127341}{4}+\frac{18435 x}{4}\right )}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx}{59535}\\ &=-\frac{558524 \sqrt{1-2 x} \sqrt{3+5 x}}{1250235 (2+3 x)^{3/2}}-\frac{1864 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{16 \int \frac{-744972-\frac{2253255 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{3750705}\\ &=-\frac{558524 \sqrt{1-2 x} \sqrt{3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{3+5 x}}{8751645 \sqrt{2+3 x}}-\frac{1864 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{32 \int \frac{-\frac{94409715}{16}-\frac{33432045 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{26254935}\\ &=-\frac{558524 \sqrt{1-2 x} \sqrt{3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{3+5 x}}{8751645 \sqrt{2+3 x}}-\frac{1864 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac{9448538 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8751645}+\frac{17830424 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{8751645}\\ &=-\frac{558524 \sqrt{1-2 x} \sqrt{3+5 x}}{1250235 (2+3 x)^{3/2}}+\frac{17830424 \sqrt{1-2 x} \sqrt{3+5 x}}{8751645 \sqrt{2+3 x}}-\frac{1864 \sqrt{1-2 x} (3+5 x)^{3/2}}{6615 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}-\frac{17830424 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645}-\frac{1717916 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8751645}\\ \end{align*}
Mathematica [A] time = 0.284285, size = 110, normalized size = 0.5 \[ \frac{8 \sqrt{2} \left (5257595 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+8915212 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (722132172 x^4+2043155529 x^3+2115318249 x^2+955601637 x+159578303\right )}{(3 x+2)^{9/2}}}{105019740} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 504, normalized size = 2.3 \begin{align*} -{\frac{2}{262549350\,{x}^{2}+26254935\,x-78764805} \left ( 722132172\,\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}+425865195\,\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}+1925685792\,\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}+1135640520\,\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}+1925685792\,\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}+1135640520\,\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}+855860352\,\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}+504729120\,\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}-21663965160\,{x}^{6}+142643392\,\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 ) +84121520\,\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 ) -63461062386\,{x}^{5}-63089824509\,{x}^{4}-16625604096\,{x}^{3}+11383710240\,{x}^{2}+8121679824\,x+1436204727 \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{3}{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 (50 \, x^{3} + 35 \, x^{2} - 12 \, 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{3}{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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