Optimal. Leaf size=122 \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}+\frac{55 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{605 \sqrt{5 x+3} \sqrt{1-2 x}}{8 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]
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Rubi [A] time = 0.0301388, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)^3}+\frac{55 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{605 \sqrt{5 x+3} \sqrt{1-2 x}}{8 (3 x+2)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx &=\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{55}{6} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{605}{8} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{605 \sqrt{1-2 x} \sqrt{3+5 x}}{8 (2+3 x)}+\frac{6655}{16} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{605 \sqrt{1-2 x} \sqrt{3+5 x}}{8 (2+3 x)}+\frac{6655}{8} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} \sqrt{3+5 x}}{12 (2+3 x)^2}+\frac{605 \sqrt{1-2 x} \sqrt{3+5 x}}{8 (2+3 x)}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{8 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0460943, size = 74, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (15707 x^2+21638 x+7488\right )}{24 (3 x+2)^3}-\frac{6655 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{336\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 539055\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1078110\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+718740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+219898\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+159720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +302932\,x\sqrt{-10\,{x}^{2}-x+3}+104832\,\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 = 3.29769, size = 144, normalized size = 1.18 \begin{align*} \frac{6655}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{27 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1043 \, \sqrt{-10 \, x^{2} - x + 3}}{108 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{15707 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06134, size = 302, normalized size = 2.48 \begin{align*} -\frac{19965 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (15707 \, x^{2} + 21638 \, x + 7488\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{336 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.86918, size = 429, normalized size = 3.52 \begin{align*} \frac{1331}{224} \, \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{1331 \,{\left (33 \, \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 )}^{5} + 11200 \, \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} + 1176000 \, \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 )}}{12 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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