Optimal. Leaf size=93 \[ \frac{4 (5 x+3)^{3/2}}{77 \sqrt{1-2 x} (3 x+2)}-\frac{29 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)}-\frac{29 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
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Rubi [A] time = 0.0216005, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{4 (5 x+3)^{3/2}}{77 \sqrt{1-2 x} (3 x+2)}-\frac{29 \sqrt{1-2 x} \sqrt{5 x+3}}{539 (3 x+2)}-\frac{29 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac{4 (3+5 x)^{3/2}}{77 \sqrt{1-2 x} (2+3 x)}+\frac{29}{77} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{29 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)}+\frac{4 (3+5 x)^{3/2}}{77 \sqrt{1-2 x} (2+3 x)}+\frac{29}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{29 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)}+\frac{4 (3+5 x)^{3/2}}{77 \sqrt{1-2 x} (2+3 x)}+\frac{29}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{29 \sqrt{1-2 x} \sqrt{3+5 x}}{539 (2+3 x)}+\frac{4 (3+5 x)^{3/2}}{77 \sqrt{1-2 x} (2+3 x)}-\frac{29 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0334624, size = 78, normalized size = 0.84 \[ \frac{7 \sqrt{5 x+3} (18 x+5)-29 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 161, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 1372+2058\,x \right ) \left ( 2\,x-1 \right ) } \left ( 174\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+29\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-58\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -252\,x\sqrt{-10\,{x}^{2}-x+3}-70\,\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 = 3.72943, size = 124, normalized size = 1.33 \begin{align*} \frac{29}{686} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{30 \, x}{49 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{19}{147 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{21 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79196, size = 239, normalized size = 2.57 \begin{align*} -\frac{29 \, \sqrt{7}{\left (6 \, x^{2} + x - 2\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 \,{\left (18 \, x + 5\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{686 \,{\left (6 \, x^{2} + x - 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.19088, size = 296, normalized size = 3.18 \begin{align*} \frac{29}{6860} \, \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{4 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{245 \,{\left (2 \, x - 1\right )}} - \frac{66 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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