Optimal. Leaf size=191 \[ -\frac{1061 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{2835}-\frac{32}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{202}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1061}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{2894 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835} \]
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Rubi [A] time = 0.0678339, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac{32}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{202}{63} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{1061}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1061 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835}-\frac{2894 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
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)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}+\frac{2}{3} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{32}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{4}{315} \int \frac{\left (\frac{4495}{4}-\frac{7575 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{202}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{32}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (\frac{1125}{2}-\frac{79575 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{4725}\\ &=-\frac{1061}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{202}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{32}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{4 \int \frac{\frac{435525}{8}+\frac{108525 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{42525}\\ &=-\frac{1061}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{202}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{32}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{2894 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2835}+\frac{11671 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5670}\\ &=-\frac{1061}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{202}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{32}{63} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2894 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835}-\frac{1061 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2835}\\ \end{align*}
Mathematica [A] time = 0.273465, size = 107, normalized size = 0.56 \[ \frac{29225 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (-2700 x^3+180 x^2+1767 x+200\right )}{\sqrt{3 x+2}}+5788 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{17010} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 150, normalized size = 0.8 \begin{align*} -{\frac{1}{510300\,{x}^{3}+391230\,{x}^{2}-119070\,x-102060}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 29225\,\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 ) +5788\,\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 ) +810000\,{x}^{5}+27000\,{x}^{4}-778500\,{x}^{3}-96810\,{x}^{2}+153030\,x+18000 \right ) } \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{3}{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}}{9 \, x^{2} + 12 \, x + 4}, 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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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