Optimal. Leaf size=123 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{\sqrt{33}}+\frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{67 \sqrt{5 x+3} \sqrt{3 x+2}}{33 \sqrt{1-2 x}}-\frac{133 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2 \sqrt{33}} \]
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Rubi [A] time = 0.0375271, antiderivative size = 123, 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, 150, 158, 113, 119} \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{67 \sqrt{5 x+3} \sqrt{3 x+2}}{33 \sqrt{1-2 x}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}-\frac{133 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{\sqrt{2+3 x} \left (\frac{37}{2}+30 x\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{67 \sqrt{2+3 x} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{-\frac{1263}{2}-\frac{1995 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{67 \sqrt{2+3 x} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}+\frac{133}{22} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx+\int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{67 \sqrt{2+3 x} \sqrt{3+5 x}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} \sqrt{3+5 x}}{3 (1-2 x)^{3/2}}-\frac{133 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2 \sqrt{33}}-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.187264, size = 115, normalized size = 0.93 \[ -\frac{-67 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} (45-167 x)+133 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{66 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 228, normalized size = 1.9 \begin{align*}{\frac{1}{66\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 134\,\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}-266\,\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}-67\,\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 ) +133\,\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 ) +5010\,{x}^{3}+4996\,{x}^{2}+294\,x-540 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \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{5}{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}}{8 \, x^{3} - 12 \, x^{2} + 6 \, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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