Optimal. Leaf size=178 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]
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Rubi [A] time = 0.0684561, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}-\frac{1}{108} \int \frac{\left (-\frac{3315}{4}-240 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{1}{648} \int \frac{\left (\frac{92115}{8}-960 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{\int \frac{\frac{2886195}{16}-33600 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{13608}\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac{200}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{3244595 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{217728}\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{3244595 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{108864}-\frac{1}{243} \left (80 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{108864 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.187706, size = 131, normalized size = 0.74 \[ \frac{-21 \sqrt{5 x+3} \left (3580650 x^4+6416859 x^3+1791504 x^2-1593212 x-677168\right )+125440 \sqrt{10-20 x} (3 x+2)^4 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-3244595 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 315, normalized size = 1.8 \begin{align*}{\frac{1}{1524096\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 262812195\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-10160640\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}+700832520\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+700832520\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+75193650\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+311481120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-12042240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+172350864\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+51913520\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2007040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +123797016\,x\sqrt{-10\,{x}^{2}-x+3}+28441056\,\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 = 3.97522, size = 266, normalized size = 1.49 \begin{align*} \frac{21775}{21168} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{95 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{168 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4355 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4704 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{539675}{42336} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{20}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3244595}{1524096} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1460395}{254016} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{18245 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28224 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62934, size = 556, normalized size = 3.12 \begin{align*} -\frac{3244595 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 ) - 125440 \, \sqrt{10}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (1790325 \, x^{3} + 4103592 \, x^{2} + 2947548 \, x + 677168\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1524096 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 3.45323, size = 602, normalized size = 3.38 \begin{align*} \frac{648919}{3048192} \, \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{20}{243} \, \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{55 \,{\left (19447 \, \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 )}^{7} + 19946472 \, \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} - 6199166400 \, \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} - 348224576000 \, \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 )}}{18144 \,{\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 )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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