Optimal. Leaf size=191 \[ -\frac{663409 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{236250}-\frac{1}{9} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{137}{315} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{9547 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{5250}-\frac{663409 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{47250}-\frac{44109377 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{472500} \]
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Rubi [A] time = 0.0653248, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ -\frac{1}{9} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac{137}{315} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{9547 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{5250}-\frac{663409 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{47250}-\frac{663409 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{236250}-\frac{44109377 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{472500} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{1}{9} \int \frac{(2+3 x)^{3/2} \sqrt{3+5 x} \left (\frac{171}{2}+137 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{137}{315} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{315} \int \frac{\left (-\frac{18135}{2}-\frac{28641 x}{2}\right ) \sqrt{2+3 x} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{9547 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{5250}-\frac{137}{315} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{2586807}{4}+\frac{1990227 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{7875}\\ &=-\frac{663409 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{47250}-\frac{9547 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{5250}-\frac{137}{315} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{\int \frac{-\frac{41887689}{2}-\frac{132328131 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{70875}\\ &=-\frac{663409 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{47250}-\frac{9547 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{5250}-\frac{137}{315} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac{7297499 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{472500}+\frac{44109377 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{472500}\\ &=-\frac{663409 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{47250}-\frac{9547 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{5250}-\frac{137}{315} \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{9} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac{44109377 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{472500}-\frac{663409 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{236250}\\ \end{align*}
Mathematica [A] time = 0.237483, size = 105, normalized size = 0.55 \[ \frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (4443376 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (236250 x^3+765000 x^2+1114065 x+1107478\right )\right )}{708750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{42525000\,{x}^{3}+32602500\,{x}^{2}-9922500\,x-8505000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -212625000\,{x}^{6}+22216880\,\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 ) -44109377\,\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 ) -851512500\,{x}^{5}-1480896000\,{x}^{4}-1562260050\,{x}^{3}-392506170\,{x}^{2}+433102080\,x+199346040 \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}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{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 (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2 \, 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{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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