Optimal. Leaf size=129 \[ \frac{412 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1323}+\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{3/2}}+\frac{412 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 \sqrt{3 x+2}}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
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Rubi [A] time = 0.037252, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{3/2}}+\frac{412 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 \sqrt{3 x+2}}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx &=\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 (2+3 x)^{3/2}}-\frac{2}{63} \int \frac{\left (-144-\frac{535 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac{412 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 (2+3 x)^{3/2}}-\frac{4 \int \frac{-\frac{10205}{4}-\frac{20785 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1323}\\ &=\frac{412 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 (2+3 x)^{3/2}}-\frac{2266 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1323}+\frac{4157 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1323}\\ &=\frac{412 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 (2+3 x)^{3/2}}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}\\ \end{align*}
Mathematica [A] time = 0.157444, size = 97, normalized size = 0.75 \[ \frac{-10955 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (723 x+475)}{(3 x+2)^{3/2}}+4157 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{3969} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 219, normalized size = 1.7 \begin{align*}{\frac{1}{39690\,{x}^{2}+3969\,x-11907} \left ( 32865\,\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}-12471\,\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}+21910\,\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 ) -8314\,\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 ) +43380\,{x}^{3}+32838\,{x}^{2}-10164\,x-8550 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,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 (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8}, 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 (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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