Optimal. Leaf size=48 \[ \frac{\sqrt{\frac{2}{33}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{\sqrt{2 x-5}} \]
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Rubi [A] time = 0.0139663, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {121, 119} \[ \frac{\sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{\sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx &=\frac{\left (\sqrt{\frac{2}{11}} \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{\sqrt{-5+2 x}}\\ &=\frac{\sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{\sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.10906, size = 79, normalized size = 1.65 \[ -\frac{\sqrt{\frac{3 x-2}{4 x+1}} (4 x+1) \sqrt{\frac{4 x-10}{44 x+11}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{11}{3}}}{\sqrt{4 x+1}}\right ),3\right )}{\sqrt{2-3 x} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 36, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{11}}{11}{\it EllipticF} \left ({\frac{2}{11}\sqrt{22-33\,x}},{\frac{i}{2}}\sqrt{2} \right ) \sqrt{5-2\,x}{\frac{1}{\sqrt{2\,x-5}}}} \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{1}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \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{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 - 3 x} \sqrt{2 x - 5} \sqrt{4 x + 1}}\, dx \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{1}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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