Optimal. Leaf size=218 \[ -\frac{3963068 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8859375 \sqrt{33}}+\frac{2}{55} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}}{1485}+\frac{4258 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{155925}+\frac{181333 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3898125}-\frac{2865161 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{19490625}-\frac{231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]
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Rubi [A] time = 0.0808639, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2}{55} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{62 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}}{1485}+\frac{4258 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{155925}+\frac{181333 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{3898125}-\frac{2865161 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{19490625}-\frac{3963068 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]
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} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx &=\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{2}{55} \int \frac{\left (-\frac{115}{2}-\frac{155 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{4 \int \frac{\left (-3145-\frac{10645 x}{4}\right ) \sqrt{1-2 x} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{7425}\\ &=\frac{4258 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{155925}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{8 \int \frac{(2+3 x)^{3/2} \left (-\frac{863165}{8}+\frac{906665 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{779625}\\ &=\frac{181333 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3898125}+\frac{4258 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{155925}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{8 \int \frac{\sqrt{2+3 x} \left (\frac{63649875}{16}+\frac{42977415 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19490625}\\ &=-\frac{2865161 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{181333 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3898125}+\frac{4258 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{155925}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{8 \int \frac{-\frac{2210338155}{16}-\frac{3465928185 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{292359375}\\ &=-\frac{2865161 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{181333 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3898125}+\frac{4258 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{155925}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{1981534 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8859375}+\frac{231061879 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{194906250}\\ &=-\frac{2865161 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{181333 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{3898125}+\frac{4258 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{155925}+\frac{62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt{3+5 x}}{1485}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}}-\frac{3963068 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.272715, size = 107, normalized size = 0.49 \[ \frac{-100280635 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4-6142500 x^3-23717250 x^2+9526995 x+7167169\right )+231061879 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{292359375 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{17541562500\,{x}^{3}+13448531250\,{x}^{2}-4093031250\,x-3508312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}+12077100000\,{x}^{6}+100280635\,\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 ) -231061879\,\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 ) -30942000000\,{x}^{5}-11093382000\,{x}^{4}+19110351150\,{x}^{3}+7213782660\,{x}^{2}-3219964590\,x-1290090420 \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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{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{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\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{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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