Optimal. Leaf size=159 \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
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Rubi [A] time = 0.0602733, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{14} \int \frac{\frac{29}{2}-120 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{1}{98} \int \frac{-\frac{2065}{4}-8505 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{514885}{8}+286125 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{11319}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{2 \int \frac{\frac{42164605}{16}-\frac{9444225 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{871563}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{4 \int \frac{2222523765}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{9587193}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{79515 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{79515 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0971551, size = 84, normalized size = 0.53 \[ \frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}}-\frac{1667076300 x^4+520073880 x^3-1053213025 x^2-169466391 x+178740084}{5478396 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 305, normalized size = 1.9 \begin{align*} -{\frac{1}{76697544\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 57150611100\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+53340570360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-25082768205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+23339068200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-28257802155\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7281034320\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2540027160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-14744982350\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3810040740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2372529474\,x\sqrt{-10\,{x}^{2}-x+3}+2502361176\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84949, size = 431, normalized size = 2.71 \begin{align*} \frac{317503395 \, \sqrt{7}{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\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 (1667076300 \, x^{4} + 520073880 \, x^{3} - 1053213025 \, x^{2} - 169466391 \, x + 178740084\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{76697544 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\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 = 2.77947, size = 479, normalized size = 3.01 \begin{align*} -\frac{15903}{38416} \, \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{625}{2662} \, \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{32 \,{\left (944 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5577 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{239679825 \,{\left (2 \, x - 1\right )}^{2}} - \frac{891 \,{\left (337 \, \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} + 75880 \, \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 )}}{4802 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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