Optimal. Leaf size=160 \[ \frac{11806 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{39375}+\frac{2}{21} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{118}{525} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{4282 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{7875}-\frac{86741 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375} \]
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Rubi [A] time = 0.0513473, 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 = {101, 154, 158, 113, 119} \[ \frac{2}{21} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{118}{525} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{4282 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{7875}+\frac{11806 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375}-\frac{86741 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375} \]
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)^{5/2} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx &=\frac{2}{21} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2}{21} \int \frac{\left (-52-\frac{177 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{118}{525} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{21} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{4 \int \frac{\left (-\frac{10809}{4}-\frac{19269 x}{4}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1575}\\ &=\frac{4282 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7875}+\frac{118}{525} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{21} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{8 \int \frac{-\frac{175761}{4}-\frac{780669 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{70875}\\ &=\frac{4282 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7875}+\frac{118}{525} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{21} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{64933 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{39375}+\frac{86741 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{39375}\\ &=\frac{4282 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{7875}+\frac{118}{525} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{2}{21} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{86741 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375}+\frac{11806 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375}\\ \end{align*}
Mathematica [A] time = 0.10712, size = 102, normalized size = 0.64 \[ \frac{-281540 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1500 x^2-3270 x+3401\right )+86741 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{118125} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 150, normalized size = 0.9 \begin{align*}{\frac{1}{3543750\,{x}^{3}+2716875\,{x}^{2}-826875\,x-708750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 281540\,\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 ) -86741\,\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 ) +1350000\,{x}^{5}-1908000\,{x}^{4}+489600\,{x}^{3}+2763390\,{x}^{2}-125610\,x-612180 \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{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, 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{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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