Optimal. Leaf size=209 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]
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Rubi [A] time = 0.0810345, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^7} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{1}{126} \int \frac{\left (-\frac{1179}{2}-1010 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{\int \frac{-\frac{394523}{4}-166690 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx}{13230}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{\int \frac{-\frac{5498457}{8}-\frac{2249265 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{370440}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{\int \frac{-\frac{73502625}{16}-\frac{7493745 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{7779240}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{958171 \sqrt{1-2 x} \sqrt{3+5 x}}{16595712 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{\int \frac{-\frac{2940587895}{32}+\frac{503039775 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{108909360}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{958171 \sqrt{1-2 x} \sqrt{3+5 x}}{16595712 (2+3 x)^2}+\frac{122343637 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{\int -\frac{149044934115}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{762365520}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{958171 \sqrt{1-2 x} \sqrt{3+5 x}}{16595712 (2+3 x)^2}+\frac{122343637 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac{52573169 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{17210368}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{958171 \sqrt{1-2 x} \sqrt{3+5 x}}{16595712 (2+3 x)^2}+\frac{122343637 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac{52573169 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{8605184}\\ &=\frac{503 \sqrt{1-2 x} \sqrt{3+5 x}}{26460 (2+3 x)^5}-\frac{149951 \sqrt{1-2 x} \sqrt{3+5 x}}{1481760 (2+3 x)^4}-\frac{71369 \sqrt{1-2 x} \sqrt{3+5 x}}{2963520 (2+3 x)^3}+\frac{958171 \sqrt{1-2 x} \sqrt{3+5 x}}{16595712 (2+3 x)^2}+\frac{122343637 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{8605184 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.204158, size = 193, normalized size = 0.92 \[ \frac{1}{42} \left (\frac{591 \sqrt{1-2 x} (5 x+3)^{7/2}}{70 (3 x+2)^5}+\frac{3 \sqrt{1-2 x} (5 x+3)^{7/2}}{(3 x+2)^6}+\frac{352839984 \sqrt{1-2 x} (5 x+3)^{7/2}-39499 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )}{21512960 (3 x+2)^4}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{1807088640\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 574887603015\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+2299550412060\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+3832584020100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+231229473930\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+3406741351200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+779215981320\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1703370675600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1049047713504\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+454232180160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+703664648512\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+50470242240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +234804461920\,x\sqrt{-10\,{x}^{2}-x+3}+31151401344\,\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 = 4.54875, size = 311, normalized size = 1.49 \begin{align*} \frac{52573169}{120472576} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{378 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{853 \, \sqrt{-10 \, x^{2} - x + 3}}{26460 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{149951 \, \sqrt{-10 \, x^{2} - x + 3}}{1481760 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{71369 \, \sqrt{-10 \, x^{2} - x + 3}}{2963520 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{958171 \, \sqrt{-10 \, x^{2} - x + 3}}{16595712 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{122343637 \, \sqrt{-10 \, x^{2} - x + 3}}{232339968 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85302, size = 517, normalized size = 2.47 \begin{align*} -\frac{788597535 \, \sqrt{7}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1807088640 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 = 5.01973, size = 676, normalized size = 3.23 \begin{align*} \frac{52573169}{1204725760} \, \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 (118497 \, \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 )}^{11} + 188015240 \, \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 )}^{9} + 122630175360 \, \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} - 17238395059200 \, \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} - 3670540357120000 \, \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} - 197895383347200000 \, \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 )}}{12907776 \,{\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 )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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