Optimal. Leaf size=187 \[ -\frac{160297 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{30250 \sqrt{33}}+\frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]
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Rubi [A] time = 0.0676765, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{665 (3 x+2)^{5/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3284 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 \sqrt{5 x+3}}-\frac{153319 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{66550}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{33} \int \frac{(2+3 x)^{5/2} \left (\frac{359}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{1}{363} \int \frac{\left (-\frac{23357}{2}-\frac{40023 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{2 \int \frac{\left (-\frac{425475}{2}-\frac{1379871 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19965}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}+\frac{2 \int \frac{\frac{60716097}{8}+\frac{47951847 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{299475}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}+\frac{160297 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{60500}+\frac{5327983 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{332750}\\ &=\frac{3284 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 \sqrt{3+5 x}}-\frac{665 (2+3 x)^{5/2}}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{153319 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{66550}-\frac{5327983 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}-\frac{160297 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{30250 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.257475, size = 102, normalized size = 0.55 \[ \frac{-5366165 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{5 \sqrt{6 x+4} \left (1078110 x^3-11321446 x^2-3117099 x+2438391\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}+10655966 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{998250 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 238, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( 3993000\,x-1996500 \right ) \left ( 30\,{x}^{3}+23\,{x}^{2}-7\,x-6 \right ) }\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10732330\,\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}-21311932\,\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}-5366165\,\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 ) +10655966\,\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 ) -32343300\,{x}^{4}+318081180\,{x}^{3}+319941890\,{x}^{2}-10809750\,x-48767820 \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 (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\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{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, 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 (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\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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