Optimal. Leaf size=158 \[ -\frac{5057 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8750}-\frac{1}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{104}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{4839 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1750}-\frac{56041 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750} \]
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Rubi [A] time = 0.0507795, antiderivative size = 158, 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} \[ -\frac{1}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{104}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{4839 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1750}-\frac{5057 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750}-\frac{56041 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750} \]
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} \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{1}{7} \int \frac{(2+3 x)^{3/2} \left (\frac{127}{2}+104 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{104}{175} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{1}{175} \int \frac{\left (-4475-\frac{14517 x}{2}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{4839 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1750}-\frac{104}{175} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{\int \frac{\frac{638619}{4}+\frac{504369 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2625}\\ &=-\frac{4839 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1750}-\frac{104}{175} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}+\frac{55627 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{17500}+\frac{168123 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{8750}\\ &=-\frac{4839 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1750}-\frac{104}{175} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1}{7} \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}-\frac{56041 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750}-\frac{5057 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750}\\ \end{align*}
Mathematica [A] time = 0.221739, size = 97, normalized size = 0.61 \[ \frac{-56455 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-5 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+6120 x+7919\right )+112082 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{8750 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 150, normalized size = 1. \begin{align*}{\frac{1}{525000\,{x}^{3}+402500\,{x}^{2}-122500\,x-105000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 56455\,\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 ) -112082\,\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 ) -675000\,{x}^{5}-2353500\,{x}^{4}-3625800\,{x}^{3}-1257970\,{x}^{2}+921530\,x+475140 \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 (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 (9 \, x^{2} + 12 \, x + 4\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{\sqrt{5 \, x + 3}{\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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