Optimal. Leaf size=120 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^2}+\frac{239 \sqrt{1-2 x} \sqrt{5 x+3}}{1764 (3 x+2)}+\frac{25}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{17687 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5292 \sqrt{7}} \]
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Rubi [A] time = 0.0426488, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 149, 157, 54, 216, 93, 204} \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^2}+\frac{239 \sqrt{1-2 x} \sqrt{5 x+3}}{1764 (3 x+2)}+\frac{25}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{17687 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5292 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac{1}{42} \int \frac{\left (-\frac{387}{2}-350 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{239 \sqrt{1-2 x} \sqrt{3+5 x}}{1764 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac{1}{882} \int \frac{-\frac{26771}{4}-12250 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{239 \sqrt{1-2 x} \sqrt{3+5 x}}{1764 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac{17687 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{10584}+\frac{125}{27} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{239 \sqrt{1-2 x} \sqrt{3+5 x}}{1764 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac{17687 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{5292}+\frac{1}{27} \left (50 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=\frac{239 \sqrt{1-2 x} \sqrt{3+5 x}}{1764 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}+\frac{25}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{17687 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{5292 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0941473, size = 94, normalized size = 0.78 \[ \frac{\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (927 x+604)}{(3 x+2)^2}-34300 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+17687 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{37044} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 191, normalized size = 1.6 \begin{align*} -{\frac{1}{74088\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 159183\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-308700\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+212244\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-411600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+70748\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -137200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -38934\,x\sqrt{-10\,{x}^{2}-x+3}-25368\,\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 = 5.32822, size = 117, normalized size = 0.98 \begin{align*} \frac{25}{54} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{17687}{74088} \, \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}}{126 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{103 \, \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 = 1.94114, size = 412, normalized size = 3.43 \begin{align*} \frac{17687 \, \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 ) - 34300 \, \sqrt{10}{\left (9 \, x^{2} + 12 \, x + 4\right )} \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 ) + 42 \,{\left (927 \, x + 604\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{74088 \,{\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 = 3.12316, size = 437, normalized size = 3.64 \begin{align*} -\frac{17687}{740880} \, \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{25}{54} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{11 \,{\left (239 \, \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} + 85400 \, \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 )}}{882 \,{\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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