Optimal. Leaf size=122 \[ \frac{4 (5 x+3)^{5/2}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac{190 (5 x+3)^{3/2}}{1617 \sqrt{1-2 x} (3 x+2)}+\frac{95 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)}+\frac{95 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]
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Rubi [A] time = 0.0287138, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{4 (5 x+3)^{5/2}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac{190 (5 x+3)^{3/2}}{1617 \sqrt{1-2 x} (3 x+2)}+\frac{95 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)}+\frac{95 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac{4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}+\frac{95}{231} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac{190 (3+5 x)^{3/2}}{1617 \sqrt{1-2 x} (2+3 x)}+\frac{4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac{95}{539} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{95 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)}+\frac{190 (3+5 x)^{3/2}}{1617 \sqrt{1-2 x} (2+3 x)}+\frac{4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac{95}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{95 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)}+\frac{190 (3+5 x)^{3/2}}{1617 \sqrt{1-2 x} (2+3 x)}+\frac{4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}-\frac{95}{343} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{95 \sqrt{1-2 x} \sqrt{3+5 x}}{3773 (2+3 x)}+\frac{190 (3+5 x)^{3/2}}{1617 \sqrt{1-2 x} (2+3 x)}+\frac{4 (3+5 x)^{5/2}}{231 (1-2 x)^{3/2} (2+3 x)}+\frac{95 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{343 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0457876, size = 86, normalized size = 0.7 \[ -\frac{7 \sqrt{5 x+3} \left (660 x^2-310 x-549\right )+285 \sqrt{7-14 x} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7203 (1-2 x)^{3/2} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 209, normalized size = 1.7 \begin{align*} -{\frac{1}{ \left ( 28812+43218\,x \right ) \left ( 2\,x-1 \right ) ^{2}} \left ( 3420\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1140\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+9240\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+570\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4340\,x\sqrt{-10\,{x}^{2}-x+3}-7686\,\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 = 2.89916, size = 163, normalized size = 1.34 \begin{align*} -\frac{95}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{550 \, x}{1029 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{20}{1029 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1825 \, x}{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{189 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{3250}{1323 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50242, size = 289, normalized size = 2.37 \begin{align*} \frac{285 \, \sqrt{7}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, 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 (660 \, x^{2} - 310 \, x - 549\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14406 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, 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 = 3.34937, size = 313, normalized size = 2.57 \begin{align*} -\frac{19}{9604} \, \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{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 )}}{343 \,{\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 )}} - \frac{2 \,{\left (116 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1023 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{25725 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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