Optimal. Leaf size=238 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{756 (3 x+2)^6}+\frac{1921 (5 x+3)^{3/2} \sqrt{1-2 x}}{1512 (3 x+2)^5}+\frac{40175505215 \sqrt{5 x+3} \sqrt{1-2 x}}{597445632 (3 x+2)}+\frac{384136145 \sqrt{5 x+3} \sqrt{1-2 x}}{42674688 (3 x+2)^2}+\frac{2199649 \sqrt{5 x+3} \sqrt{1-2 x}}{1524096 (3 x+2)^3}-\frac{443563 \sqrt{5 x+3} \sqrt{1-2 x}}{254016 (3 x+2)^4}-\frac{1891543995 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2458624 \sqrt{7}} \]
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Rubi [A] time = 0.0957046, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{21 (3 x+2)^7}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{756 (3 x+2)^6}+\frac{1921 (5 x+3)^{3/2} \sqrt{1-2 x}}{1512 (3 x+2)^5}+\frac{40175505215 \sqrt{5 x+3} \sqrt{1-2 x}}{597445632 (3 x+2)}+\frac{384136145 \sqrt{5 x+3} \sqrt{1-2 x}}{42674688 (3 x+2)^2}+\frac{2199649 \sqrt{5 x+3} \sqrt{1-2 x}}{1524096 (3 x+2)^3}-\frac{443563 \sqrt{5 x+3} \sqrt{1-2 x}}{254016 (3 x+2)^4}-\frac{1891543995 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2458624 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}-\frac{1}{378} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x} \left (-\frac{6285}{4}+1245 x\right )}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{\int \frac{\left (\frac{1131615}{8}-\frac{407325 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx}{5670}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{\int \frac{\frac{38989515}{16}-\frac{14249325 x}{4}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{476280}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{\int \frac{\frac{7285747875}{32}-\frac{1154815725 x}{4}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{10001880}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}+\frac{384136145 \sqrt{1-2 x} \sqrt{3+5 x}}{42674688 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{\int \frac{\frac{868352079525}{64}-\frac{201671476125 x}{16}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{140026320}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}+\frac{384136145 \sqrt{1-2 x} \sqrt{3+5 x}}{42674688 (2+3 x)^2}+\frac{40175505215 \sqrt{1-2 x} \sqrt{3+5 x}}{597445632 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{\int \frac{48262745032425}{128 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{980184240}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}+\frac{384136145 \sqrt{1-2 x} \sqrt{3+5 x}}{42674688 (2+3 x)^2}+\frac{40175505215 \sqrt{1-2 x} \sqrt{3+5 x}}{597445632 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{1891543995 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{4917248}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}+\frac{384136145 \sqrt{1-2 x} \sqrt{3+5 x}}{42674688 (2+3 x)^2}+\frac{40175505215 \sqrt{1-2 x} \sqrt{3+5 x}}{597445632 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}+\frac{1891543995 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2458624}\\ &=-\frac{443563 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)^4}+\frac{2199649 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)^3}+\frac{384136145 \sqrt{1-2 x} \sqrt{3+5 x}}{42674688 (2+3 x)^2}+\frac{40175505215 \sqrt{1-2 x} \sqrt{3+5 x}}{597445632 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{756 (2+3 x)^6}+\frac{1921 \sqrt{1-2 x} (3+5 x)^{3/2}}{1512 (2+3 x)^5}-\frac{1891543995 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2458624 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.302452, size = 221, normalized size = 0.93 \[ \frac{1}{49} \left (\frac{47 (5 x+3)^{5/2} (1-2 x)^{7/2}}{4 (3 x+2)^6}+\frac{3 (5 x+3)^{5/2} (1-2 x)^{7/2}}{(3 x+2)^7}+\frac{783 \left (43904 (1-2 x)^{5/2} (5 x+3)^{5/2}+55 (3 x+2) \left (5488 (1-2 x)^{3/2} (5 x+3)^{5/2}+11 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}-11 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )\right )}{351232 (3 x+2)^5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 394, normalized size = 1.7 \begin{align*}{\frac{1}{34420736\, \left ( 2+3\,x \right ) ^{7}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4136806717065\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}+19305098012970\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+38610196025940\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+1687371219030\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+42900217806600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+6824775560540\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+28600145204400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+11504134299504\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+11440058081760\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+10344747708288\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2542235129280\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5233883952416\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+242117631360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1412695104576\,x\sqrt{-10\,{x}^{2}-x+3}+158916941568\,\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 = 1.82092, size = 437, normalized size = 1.84 \begin{align*} \frac{118356975}{4302592} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{7 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{305 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{588 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{2161 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1176 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{129195 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{4780215 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{307328 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{213042555 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8605184 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{2892030075}{8605184} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1891543995}{34420736} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2548112985}{17210368} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{280970415 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{17210368 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91495, size = 598, normalized size = 2.51 \begin{align*} -\frac{1891543995 \, \sqrt{7}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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 (120526515645 \, x^{6} + 487483968610 \, x^{5} + 821723878536 \, x^{4} + 738910550592 \, x^{3} + 373848853744 \, x^{2} + 100906793184 \, x + 11351210112\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{34420736 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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 = 6.31948, size = 759, normalized size = 3.19 \begin{align*} \frac{378308799}{68841472} \, \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{805255 \,{\left (2349 \, \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 )}^{13} + 4384800 \, \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} - 4393081280 \, \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} - 1503513804800 \, \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} - 272402016768000 \, \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} - 26951436288000000 \, \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} - 1131960324096000000 \, \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 )}}{1229312 \,{\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 )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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