Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]
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Rubi [A] time = 0.0163254, 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} \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{107}{40} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{107}{80} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{1177}{160} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{107}{80} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{1177 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{80 \sqrt{5}}\\ &=-\frac{107}{80} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{80 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0220809, size = 55, normalized size = 0.76 \[ \frac{1}{800} \left (-10 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+143)-1177 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 70, normalized size = 1. \begin{align*}{\frac{1}{1600}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1177\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1200\,x\sqrt{-10\,{x}^{2}-x+3}-2860\,\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 = 3.67312, size = 59, normalized size = 0.82 \begin{align*} \frac{1177}{1600} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{143}{80} \, \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 = 1.78296, size = 201, normalized size = 2.79 \begin{align*} -\frac{1}{80} \,{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1177}{1600} \, \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 = 7.6089, size = 167, normalized size = 2.32 \begin{align*} \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.0783, size = 61, normalized size = 0.85 \begin{align*} -\frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 1177 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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