Optimal. Leaf size=122 \[ \frac{23 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0332651, antiderivative size = 122, 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 = {97, 154, 158, 113, 119} \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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)^{3/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\int \frac{\sqrt{2+3 x} \left (\frac{37}{2}+30 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=2 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{1}{15} \int \frac{-660-\frac{2085 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=2 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{23}{10} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{139}{10} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=2 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.17441, size = 103, normalized size = 0.84 \[ \frac{70 \sqrt{2-4 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-30 \sqrt{3 x+2} \sqrt{5 x+3} (x-4)-139 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 140, normalized size = 1.2 \begin{align*} -{\frac{1}{900\,{x}^{3}+690\,{x}^{2}-210\,x-180}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\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 ) -139\,\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 ) -450\,{x}^{3}+1230\,{x}^{2}+2100\,x+720 \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{3}{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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{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{3}{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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