Optimal. Leaf size=180 \[ \frac{87374783 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{835409 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{293 \sqrt{1-2 x} \sqrt{5 x+3}}{120 (3 x+2)^4}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{333216939 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]
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Rubi [A] time = 0.06689, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ \frac{87374783 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{835409 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{293 \sqrt{1-2 x} \sqrt{5 x+3}}{120 (3 x+2)^4}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{333216939 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^6 \sqrt{3+5 x}} \, dx &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{\frac{337}{2}-260 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{1}{420} \int \frac{\frac{85323}{4}-30765 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{\int \frac{\frac{15850275}{8}-2510445 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{8820}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{835409 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{\int \frac{\frac{1888544805}{16}-\frac{438589725 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{123480}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{835409 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{87374783 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{\int \frac{104963335785}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{864360}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{835409 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{87374783 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{333216939 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{835409 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{87374783 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{333216939 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=\frac{7 \sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{293 \sqrt{1-2 x} \sqrt{3+5 x}}{120 (2+3 x)^4}+\frac{23909 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{835409 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{87374783 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}-\frac{333216939 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.129285, size = 135, normalized size = 0.75 \[ \frac{1}{35} \left (\frac{5 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (262124349 x^2+361165738 x+124968544\right )}{(3 x+2)^3}-333216939 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{43904}+\frac{975 \sqrt{5 x+3} (1-2 x)^{5/2}}{56 (3 x+2)^4}+\frac{3 \sqrt{5 x+3} (1-2 x)^{5/2}}{(3 x+2)^5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 404858580885\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+1349528602950\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+1799371470600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+165138339870\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1199580980400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+447737213700\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+399860326800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+455499158856\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+53314710240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +206091285904\,x\sqrt{-10\,{x}^{2}-x+3}+34994513344\,\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 = 2.72956, size = 248, normalized size = 1.38 \begin{align*} \frac{333216939}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{293 \, \sqrt{-10 \, x^{2} - x + 3}}{120 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{23909 \, \sqrt{-10 \, x^{2} - x + 3}}{1680 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{835409 \, \sqrt{-10 \, x^{2} - x + 3}}{9408 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{87374783 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58555, size = 455, normalized size = 2.53 \begin{align*} -\frac{1666084695 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (11795595705 \, x^{4} + 31981229550 \, x^{3} + 32535654204 \, x^{2} + 14720806136 \, x + 2499608096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 3.53892, size = 594, normalized size = 3.3 \begin{align*} \frac{333216939}{6146560} \, \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{121 \,{\left (8222141 \, \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} + 5797080240 \, \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} + 1842336276480 \, \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} + 282112659584000 \, \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} + 16926759575040000 \, \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 )}}{21952 \,{\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 )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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