Optimal. Leaf size=104 \[ \frac{3 (1-2 x)^{5/2}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{5 x+3}}-169 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0281761, antiderivative size = 104, 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} \[ \frac{3 (1-2 x)^{5/2}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{5 x+3}}-169 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac{3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac{169}{14} \int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}-\frac{169}{2} \int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{3+5 x}}+\frac{1183}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{3+5 x}}+1183 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{169 (1-2 x)^{3/2}}{21 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{5/2}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{3+5 x}}-169 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.038643, size = 92, normalized size = 0.88 \[ \frac{\sqrt{1-2 x} \left (7755 x^2+9652 x+2995\right )-507 \sqrt{7} \sqrt{5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3 (3 x+2) (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 202, normalized size = 1.9 \begin{align*}{\frac{1}{12+18\,x} \left ( 38025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+70980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+44109\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+15510\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+9126\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +19304\,x\sqrt{-10\,{x}^{2}-x+3}+5990\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.63941, size = 163, normalized size = 1.57 \begin{align*} \frac{169}{2} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1034 \, x}{3 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2699}{15 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3902 \, x}{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{343}{27 \,{\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{6343}{135 \,{\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 = 1.56717, size = 298, normalized size = 2.87 \begin{align*} -\frac{507 \, \sqrt{7}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 ) - 2 \,{\left (7755 \, x^{2} + 9652 \, x + 2995\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 = 2.81797, size = 423, normalized size = 4.07 \begin{align*} -\frac{1}{240} \, \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} + \frac{169}{20} \, \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{34}{5} \, \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 )} + \frac{462 \, \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 )}}{{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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