Optimal. Leaf size=155 \[ \frac{4817 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{250 \sqrt{33}}+\frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{419}{50} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{7279}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0458242, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\sqrt{5 x+3} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{419}{50} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{4817 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250 \sqrt{33}}+\frac{7279}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^{3/2} \left (\frac{55}{2}+45 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{9}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{1}{25} \int \frac{\left (-\frac{3875}{2}-\frac{6285 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{419}{50} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{9}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{1}{375} \int \frac{\frac{276495}{4}+109185 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{419}{50} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{9}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{4817}{500} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{7279}{125} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{419}{50} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{9}{5} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{(2+3 x)^{5/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{7279}{125} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4817 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.198394, size = 110, normalized size = 0.71 \[ \frac{14665 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+328 x-799\right )-29116 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1500 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{45000\,{x}^{3}+34500\,{x}^{2}-10500\,x-9000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 14665\,\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 ) -29116\,\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 ) -40500\,{x}^{4}-198900\,{x}^{3}+156390\,{x}^{2}+396390\,x+143820 \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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\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 (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, 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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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