Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]
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Rubi [A] time = 0.0154829, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx &=-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{41}{40} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451}{400} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{200 \sqrt{5}}\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{200 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0280298, size = 64, normalized size = 0.89 \[ \frac{10 \sqrt{5 x+3} \left (-120 x^2+38 x+11\right )-451 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 70, normalized size = 1. \begin{align*}{\frac{1}{4000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 451\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1200\,x\sqrt{-10\,{x}^{2}-x+3}+220\,\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 = 3.26988, size = 59, normalized size = 0.82 \begin{align*} \frac{451}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{10} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{11}{200} \, \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 = 1.37428, size = 198, normalized size = 2.75 \begin{align*} \frac{1}{200} \,{\left (60 \, x + 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{451}{4000} \, \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.14694, size = 165, normalized size = 2.29 \begin{align*} - \frac{7 \sqrt{2} \left (\begin{cases} \frac{11 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.97485, size = 116, normalized size = 1.61 \begin{align*} \frac{3}{2000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \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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