Optimal. Leaf size=122 \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]
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Rubi [A] time = 0.029413, 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} \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{11}{6} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{121}{56} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{1331}{784} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{1331}{392} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{392 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0447017, size = 74, normalized size = 0.61 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4223 x^2+4478 x+1152\right )}{(3 x+2)^3}-3993 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8232} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{16464\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 107811\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+215622\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+143748\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+59122\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+31944\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +62692\,x\sqrt{-10\,{x}^{2}-x+3}+16128\,\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 = 1.75043, size = 163, normalized size = 1.34 \begin{align*} \frac{1331}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{55}{294} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{407 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80625, size = 301, normalized size = 2.47 \begin{align*} -\frac{3993 \, \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 (4223 \, x^{2} + 4478 \, x + 1152\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{16464 \,{\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 = 2.5226, size = 429, normalized size = 3.52 \begin{align*} \frac{1331}{54880} \, \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 (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 )}^{5} + 2240 \, \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} - 235200 \, \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 )}}{588 \,{\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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