Optimal. Leaf size=238 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{756 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac{12421 \sqrt{1-2 x} (5 x+3)^{3/2}}{52920 (3 x+2)^5}+\frac{23466191827 \sqrt{1-2 x} \sqrt{5 x+3}}{4182119424 (3 x+2)}+\frac{224018941 \sqrt{1-2 x} \sqrt{5 x+3}}{298722816 (3 x+2)^2}+\frac{6249601 \sqrt{1-2 x} \sqrt{5 x+3}}{53343360 (3 x+2)^3}-\frac{1289227 \sqrt{1-2 x} \sqrt{5 x+3}}{8890560 (3 x+2)^4}-\frac{1104970911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{17210368 \sqrt{7}} \]
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Rubi [A] time = 0.0960751, 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} \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{756 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac{12421 \sqrt{1-2 x} (5 x+3)^{3/2}}{52920 (3 x+2)^5}+\frac{23466191827 \sqrt{1-2 x} \sqrt{5 x+3}}{4182119424 (3 x+2)}+\frac{224018941 \sqrt{1-2 x} \sqrt{5 x+3}}{298722816 (3 x+2)^2}+\frac{6249601 \sqrt{1-2 x} \sqrt{5 x+3}}{53343360 (3 x+2)^3}-\frac{1289227 \sqrt{1-2 x} \sqrt{5 x+3}}{8890560 (3 x+2)^4}-\frac{1104970911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{17210368 \sqrt{7}} \]
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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{1}{378} \int \frac{(3+5 x)^{3/2} \left (-\frac{6461}{4}+2235 x\right )}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{\int \frac{\sqrt{3+5 x} \left (-\frac{693747}{8}+\frac{223305 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^5} \, dx}{39690}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{\int \frac{-\frac{31720047}{16}+\frac{4510185 x}{4}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{3333960}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{\int \frac{-\frac{4340886375}{32}+\frac{656208105 x}{4}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{70013160}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}+\frac{224018941 \sqrt{1-2 x} \sqrt{3+5 x}}{298722816 (2+3 x)^2}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{\int \frac{-\frac{507690196545}{64}+\frac{117609944025 x}{16}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{980184240}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}+\frac{224018941 \sqrt{1-2 x} \sqrt{3+5 x}}{298722816 (2+3 x)^2}+\frac{23466191827 \sqrt{1-2 x} \sqrt{3+5 x}}{4182119424 (2+3 x)}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{\int -\frac{28193332794165}{128 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6861289680}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}+\frac{224018941 \sqrt{1-2 x} \sqrt{3+5 x}}{298722816 (2+3 x)^2}+\frac{23466191827 \sqrt{1-2 x} \sqrt{3+5 x}}{4182119424 (2+3 x)}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}+\frac{1104970911 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{34420736}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}+\frac{224018941 \sqrt{1-2 x} \sqrt{3+5 x}}{298722816 (2+3 x)^2}+\frac{23466191827 \sqrt{1-2 x} \sqrt{3+5 x}}{4182119424 (2+3 x)}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}+\frac{1104970911 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{17210368}\\ &=-\frac{1289227 \sqrt{1-2 x} \sqrt{3+5 x}}{8890560 (2+3 x)^4}+\frac{6249601 \sqrt{1-2 x} \sqrt{3+5 x}}{53343360 (2+3 x)^3}+\frac{224018941 \sqrt{1-2 x} \sqrt{3+5 x}}{298722816 (2+3 x)^2}+\frac{23466191827 \sqrt{1-2 x} \sqrt{3+5 x}}{4182119424 (2+3 x)}-\frac{12421 \sqrt{1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac{1104970911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{17210368 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.295526, size = 221, normalized size = 0.93 \[ \frac{1}{49} \left (\frac{263 (1-2 x)^{5/2} (5 x+3)^{7/2}}{28 (3 x+2)^6}+\frac{3 (1-2 x)^{5/2} (5 x+3)^{7/2}}{(3 x+2)^7}+\frac{2287 \left (307328 (1-2 x)^{3/2} (5 x+3)^{7/2}+11 (3 x+2) \left (115248 \sqrt{1-2 x} (5 x+3)^{7/2}-11 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )\right )}{12293120 (3 x+2)^5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 394, normalized size = 1.7 \begin{align*}{\frac{1}{1204725760\, \left ( 2+3\,x \right ) ^{7}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 12082856911785\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}+56386665588330\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+112773331176660\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+4927900283670\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+125303701307400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+19931139696860\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+83535800871600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+33595896368496\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+33414320348640\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+30215645552512\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7425404521920\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+15290511878432\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+707181383040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4126308877376\,x\sqrt{-10\,{x}^{2}-x+3}+463681177344\,\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 = 1.86469, size = 437, normalized size = 1.84 \begin{align*} \frac{207419465}{90354432} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{49 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{157 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4116 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{6289 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{41160 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{75471 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{153664 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{2792427 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2151296 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{124451679 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{60236288 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1689418335}{60236288} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1104970911}{240945152} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1488514533}{120472576} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{492397961 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{361417728 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62171, size = 606, normalized size = 2.55 \begin{align*} -\frac{5524854555 \, \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 (351992877405 \, x^{6} + 1423652835490 \, x^{5} + 2399706883464 \, x^{4} + 2158260396608 \, x^{3} + 1092179419888 \, x^{2} + 294736348384 \, x + 33120084096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1204725760 \,{\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 = 5.80444, size = 759, normalized size = 3.19 \begin{align*} \frac{1104970911}{2409451520} \, \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{161051 \,{\left (6861 \, \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} + 12807200 \, \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} + 10148425280 \, \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} - 3461100339200 \, \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} - 785566018048000 \, \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} - 78720223232000000 \, \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} - 3306249375744000000 \, \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 )}}{8605184 \,{\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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