Optimal. Leaf size=57 \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
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Rubi [A] time = 0.0137494, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{1}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x}}-\frac{2}{7} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{7 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0230921, size = 57, normalized size = 1. \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 108, normalized size = 1.9 \begin{align*} -{\frac{1}{98\,x-49} \left ( 2\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-\sqrt{7}\arctan \left ({\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \right ) +14\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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 = 2.93576, size = 78, normalized size = 1.37 \begin{align*} -\frac{1}{49} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{10 \, x}{7 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6}{7 \, \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 = 1.70992, size = 201, normalized size = 3.53 \begin{align*} \frac{\sqrt{7}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{49 \,{\left (2 \, x - 1\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{\sqrt{5 x + 3}}{\left (1 - 2 x\right )^{\frac{3}{2}} \left (3 x + 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.28974, size = 135, normalized size = 2.37 \begin{align*} -\frac{1}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{35 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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