Optimal. Leaf size=129 \[ -\frac{598 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5625}+\frac{2}{25} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{194 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1125}-\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5625} \]
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Rubi [A] time = 0.0392476, antiderivative size = 129, 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 = {101, 154, 158, 113, 119} \[ \frac{2}{25} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{194 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1125}-\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5625}-\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx &=\frac{2}{25} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2}{25} \int \frac{\left (-\frac{67}{2}-\frac{97 x}{2}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1125}+\frac{2}{25} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{4 \int \frac{-584-\frac{2797 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1125}\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1125}+\frac{2}{25} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2797 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5625}+\frac{3289 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5625}\\ &=\frac{194 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1125}+\frac{2}{25} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5625}-\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5625}\\ \end{align*}
Mathematica [A] time = 0.185345, size = 97, normalized size = 0.75 \[ \frac{7070 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (71-45 x)+2797 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{16875} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 145, normalized size = 1.1 \begin{align*} -{\frac{1}{506250\,{x}^{3}+388125\,{x}^{2}-118125\,x-101250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 7070\,\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 ) +2797\,\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 ) +81000\,{x}^{4}-65700\,{x}^{3}-116880\,{x}^{2}+13620\,x+25560 \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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}}\,{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{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}}, 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{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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