Optimal. Leaf size=50 \[ \frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]
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Rubi [A] time = 0.0097355, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ \frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx &=\frac{1}{5} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{10} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{1}{5} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5 \sqrt{5}}\\ &=\frac{1}{5} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0206144, size = 59, normalized size = 1.18 \[ \frac{10 (1-2 x) \sqrt{5 x+3}-11 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{50 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 56, normalized size = 1.1 \begin{align*}{\frac{1}{5}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{11\,\sqrt{10}}{100}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.1448, size = 39, normalized size = 0.78 \begin{align*} \frac{11}{100} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{5} \, \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 = 1.18888, size = 178, normalized size = 3.56 \begin{align*} -\frac{11}{100} \, \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{1}{5} \, \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 [A] time = 1.61473, size = 141, normalized size = 2.82 \begin{align*} \begin{cases} \frac{2 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{10 x - 5}} - \frac{11 i \sqrt{x + \frac{3}{5}}}{5 \sqrt{10 x - 5}} - \frac{11 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{50} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{50} - \frac{2 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{5 - 10 x}} + \frac{11 \sqrt{x + \frac{3}{5}}}{5 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.69097, size = 54, normalized size = 1.08 \begin{align*} \frac{1}{50} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \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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