Optimal. Leaf size=98 \[ \frac{1}{7} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{34}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0272268, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {98, 158, 113, 119} \[ \frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{1}{7} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{34}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} \sqrt{2+3 x}} \, dx &=\frac{11 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{1}{7} \int \frac{\frac{215}{2}+170 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{11}{14} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{34}{7} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{11 \sqrt{2+3 x} \sqrt{3+5 x}}{7 \sqrt{1-2 x}}+\frac{34}{7} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1}{7} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.0797762, size = 92, normalized size = 0.94 \[ \frac{1}{42} \left (35 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{66 \sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}-68 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 135, normalized size = 1.4 \begin{align*} -{\frac{1}{1260\,{x}^{3}+966\,{x}^{2}-294\,x-252}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\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 ) -68\,\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 ) +990\,{x}^{2}+1254\,x+396 \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{3}{2}}}{\sqrt{3 \, x + 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 (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2}, 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{3}{2}}}{\sqrt{3 \, x + 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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