Optimal. Leaf size=222 \[ -\frac{32836 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{109375}-\frac{284}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{22866 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}+\frac{49321 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]
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Rubi [A] time = 0.0796846, 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} \[ -\frac{284}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{22866 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{32836 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}+\frac{49321 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]
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} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (-\frac{5}{2}-30 x\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (-15-\frac{3195 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}-\frac{284}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{134235}{4}-\frac{514485 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{7875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}+\frac{22866 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{284}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{8 \int \frac{\sqrt{2+3 x} \left (-\frac{561375}{8}+\frac{2280015 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{196875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}+\frac{33778 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{22866 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{284}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{8 \int \frac{\frac{881145}{8}-\frac{6658335 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2953125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}+\frac{33778 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{22866 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{284}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{49321 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{109375}+\frac{180598 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{109375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt{3+5 x}}+\frac{33778 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{21875}+\frac{22866 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{4375}-\frac{284}{175} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{49321 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}-\frac{32836 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}\\ \end{align*}
Mathematica [A] time = 0.286523, size = 112, normalized size = 0.5 \[ \frac{591115 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (67500 x^4-47250 x^3-41025 x^2-23425 x-19087\right )}{(5 x+3)^{3/2}}-49321 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{328125} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 234, normalized size = 1.1 \begin{align*} -{\frac{1}{1968750\,{x}^{2}+328125\,x-656250} \left ( 2955575\,\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}-246605\,\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}-4050000\,{x}^{6}+1773345\,\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 ) -147963\,\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 ) +2160000\,{x}^{5}+4284000\,{x}^{4}+870750\,{x}^{3}+558970\,{x}^{2}-277630\,x-381740 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{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 (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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 (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, 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 (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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