Optimal. Leaf size=93 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac{33 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
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Rubi [A] time = 0.020884, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac{33 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac{33}{28} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac{363}{392} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac{363}{196} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{196 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0365153, size = 69, normalized size = 0.74 \[ \frac{-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)}{(3 x+2)^2}-363 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 154, normalized size = 1.7 \begin{align*}{\frac{1}{2744\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4356\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1452\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2366\,x\sqrt{-10\,{x}^{2}-x+3}-1512\,\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 = 4.16567, size = 103, normalized size = 1.11 \begin{align*} \frac{363}{2744} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{42 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{169 \, \sqrt{-10 \, x^{2} - x + 3}}{588 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67853, size = 254, normalized size = 2.73 \begin{align*} -\frac{363 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\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 (169 \, x + 108\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2744 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 2.37965, size = 347, normalized size = 3.73 \begin{align*} \frac{363}{27440} \, \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{121 \,{\left (3 \, \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 )}^{3} + 1400 \, \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 )}\right )}}{98 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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