Optimal. Leaf size=191 \[ -\frac{76163 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1063125}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{4725}-\frac{839 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{23625}-\frac{76163 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{212625}-\frac{4971289 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2126250} \]
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Rubi [A] time = 0.068456, 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} \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{4725}-\frac{839 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{23625}-\frac{76163 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{212625}-\frac{76163 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1063125}-\frac{4971289 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2126250} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx &=\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{2}{45} \int \frac{\left (-\frac{67}{2}-\frac{97 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{4725}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{\left (-619-\frac{2517 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{4725}\\ &=-\frac{839 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{4725}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{290799}{8}+\frac{228489 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{70875}\\ &=-\frac{76163 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{212625}-\frac{839 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{4725}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4 \int \frac{-\frac{2362749}{2}-\frac{14913867 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{637875}\\ &=-\frac{76163 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{212625}-\frac{839 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{4725}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{837793 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2126250}+\frac{4971289 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2126250}\\ &=-\frac{76163 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{212625}-\frac{839 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{23625}+\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{4725}+\frac{2}{45} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{4971289 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2126250}-\frac{76163 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1063125}\\ \end{align*}
Mathematica [A] time = 0.242514, size = 105, normalized size = 0.55 \[ \frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (491582 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+112500 x^2-337545 x-64804\right )\right )}{3189375 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{191362500\,{x}^{3}+146711250\,{x}^{2}-44651250\,x-38272500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -425250000\,{x}^{6}+2457910\,\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 ) -4971289\,\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 ) -427275000\,{x}^{5}+325390500\,{x}^{4}+399904650\,{x}^{3}-5919690\,{x}^{2}-74366940\,x-11664720 \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{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 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 (-{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-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{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 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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