Optimal. Leaf size=50 \[ \frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}}-\frac{3}{10} \sqrt{1-2 x} \sqrt{5 x+3} \]
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Rubi [A] time = 0.0101219, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 54, 216} \[ \frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}}-\frac{3}{10} \sqrt{1-2 x} \sqrt{5 x+3} \]
Antiderivative was successfully verified.
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Rule 80
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{2+3 x}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{3}{10} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{37}{20} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{3}{10} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{37 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{10 \sqrt{5}}\\ &=-\frac{3}{10} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{10 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.012048, size = 50, normalized size = 1. \[ -\frac{3}{10} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{37 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{10 \sqrt{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 1.1 \begin{align*}{\frac{1}{200}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 37\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -60\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.93875, size = 35, normalized size = 0.7 \begin{align*} -\frac{37}{200} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{3}{10} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75673, size = 180, normalized size = 3.6 \begin{align*} -\frac{37}{200} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac{3}{10} \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} \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{3 x + 2}{\sqrt{1 - 2 x} \sqrt{5 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.32729, size = 54, normalized size = 1.08 \begin{align*} \frac{1}{100} \, \sqrt{5}{\left (37 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - 6 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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