Optimal. Leaf size=209 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{152571047 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)}+\frac{1460201 \sqrt{1-2 x} \sqrt{5 x+3}}{2370816 (3 x+2)^2}+\frac{42461 \sqrt{1-2 x} \sqrt{5 x+3}}{423360 (3 x+2)^3}+\frac{4619 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{3780 (3 x+2)^5}-\frac{64645339 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]
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Rubi [A] time = 0.0815376, 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 = {97, 149, 151, 12, 93, 204} \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{152571047 \sqrt{1-2 x} \sqrt{5 x+3}}{33191424 (3 x+2)}+\frac{1460201 \sqrt{1-2 x} \sqrt{5 x+3}}{2370816 (3 x+2)^2}+\frac{42461 \sqrt{1-2 x} \sqrt{5 x+3}}{423360 (3 x+2)^3}+\frac{4619 \sqrt{1-2 x} \sqrt{5 x+3}}{211680 (3 x+2)^4}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{3780 (3 x+2)^5}-\frac{64645339 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]
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{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{\left (\frac{9}{2}-20 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{\int \frac{-\frac{2087}{4}-1360 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx}{1890}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{\int \frac{\frac{112467}{8}-\frac{69285 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{52920}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{\int \frac{\frac{27328875}{16}-\frac{4458405 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{1111320}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}+\frac{1460201 \sqrt{1-2 x} \sqrt{3+5 x}}{2370816 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{\int \frac{\frac{3295705245}{32}-\frac{766605525 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{15558480}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}+\frac{1460201 \sqrt{1-2 x} \sqrt{3+5 x}}{2370816 (2+3 x)^2}+\frac{152571047 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{\int \frac{183269536065}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{108909360}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}+\frac{1460201 \sqrt{1-2 x} \sqrt{3+5 x}}{2370816 (2+3 x)^2}+\frac{152571047 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{64645339 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2458624}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}+\frac{1460201 \sqrt{1-2 x} \sqrt{3+5 x}}{2370816 (2+3 x)^2}+\frac{152571047 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{64645339 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1229312}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{3780 (2+3 x)^5}+\frac{4619 \sqrt{1-2 x} \sqrt{3+5 x}}{211680 (2+3 x)^4}+\frac{42461 \sqrt{1-2 x} \sqrt{3+5 x}}{423360 (2+3 x)^3}+\frac{1460201 \sqrt{1-2 x} \sqrt{3+5 x}}{2370816 (2+3 x)^2}+\frac{152571047 \sqrt{1-2 x} \sqrt{3+5 x}}{33191424 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}-\frac{64645339 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1229312 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.137663, size = 171, normalized size = 0.82 \[ \frac{1}{42} \left (\frac{48569 \left (7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4223 x^2+4478 x+1152\right )-3993 \sqrt{7} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{614656 (3 x+2)^3}+\frac{20103 (1-2 x)^{3/2} (5 x+3)^{5/2}}{560 (3 x+2)^4}+\frac{789 (1-2 x)^{3/2} (5 x+3)^{5/2}}{70 (3 x+2)^5}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^6}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{258155520\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 706896781965\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+2827587127860\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+4712645213100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+288359278830\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+4189017967200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+974076568920\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+2094508983600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1316639800224\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+558535728960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+890190644672\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+62059525440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +300969324320\,x\sqrt{-10\,{x}^{2}-x+3}+40689258624\,\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 = 1.78308, size = 329, normalized size = 1.57 \begin{align*} \frac{64645339}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2671295}{921984} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{42 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{29 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{980 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{1273 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{7840 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{45245 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{65856 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1602777 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{19767583 \, \sqrt{-10 \, x^{2} - x + 3}}{3687936 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06619, size = 516, normalized size = 2.47 \begin{align*} -\frac{969680085 \, \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 (20597091345 \, x^{5} + 69576897780 \, x^{4} + 94045700016 \, x^{3} + 63585046048 \, x^{2} + 21497808880 \, x + 2906375616\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{258155520 \,{\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 = 4.65358, size = 676, normalized size = 3.23 \begin{align*} \frac{64645339}{172103680} \, \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 (145707 \, \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} + 231188440 \, \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} - 144245619840 \, \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} - 41365512115200 \, \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} - 5067855403520000 \, \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} - 250767109017600000 \, \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 )}}{1843968 \,{\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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