Optimal. Leaf size=160 \[ -\frac{178 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1029}-\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 \sqrt{3 x+2}}-\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}+\frac{458 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029} \]
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Rubi [A] time = 0.0501185, antiderivative size = 160, 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 = {98, 152, 158, 113, 119} \[ -\frac{458 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 \sqrt{3 x+2}}-\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{3/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{178 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029}+\frac{458 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029} \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{1}{7} \int \frac{-\frac{181}{2}-160 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{3/2}}-\frac{2}{147} \int \frac{-\frac{247}{2}-\frac{485 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 \sqrt{2+3 x}}-\frac{4 \int \frac{\frac{395}{4}+\frac{1145 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1029}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 \sqrt{2+3 x}}-\frac{458 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1029}+\frac{979 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1029}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{147 (2+3 x)^{3/2}}-\frac{458 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 \sqrt{2+3 x}}+\frac{458 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029}-\frac{178 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1029}\\ \end{align*}
Mathematica [A] time = 0.206977, size = 97, normalized size = 0.61 \[ \frac{\sqrt{2} \left (3395 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{3 \sqrt{10 x+6} \left (1374 x^2+908 x+11\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}-458 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3087} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 219, normalized size = 1.4 \begin{align*} -{\frac{1}{30870\,{x}^{2}+3087\,x-9261}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 10185\,\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}-1374\,\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}+6790\,\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 ) -916\,\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 ) +41220\,{x}^{3}+51972\,{x}^{2}+16674\,x+198 \right ) \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{3}{2}}}{{\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 (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, 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{3}{2}}}{{\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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