Optimal. Leaf size=180 \[ \frac{17 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac{187 \sqrt{1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac{2057 \sqrt{1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac{22627 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{248897 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]
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Rubi [A] time = 0.0556498, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{17 \sqrt{1-2 x} (5 x+3)^{7/2}}{40 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}-\frac{187 \sqrt{1-2 x} (5 x+3)^{5/2}}{1680 (3 x+2)^3}-\frac{2057 \sqrt{1-2 x} (5 x+3)^{3/2}}{9408 (3 x+2)^2}-\frac{22627 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{248897 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17}{10} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{187}{80} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{187 \sqrt{1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{2057}{672} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{2057 \sqrt{1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac{187 \sqrt{1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{22627 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{6272}\\ &=-\frac{22627 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{2057 \sqrt{1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac{187 \sqrt{1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{248897 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=-\frac{22627 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{2057 \sqrt{1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac{187 \sqrt{1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}+\frac{248897 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=-\frac{22627 \sqrt{1-2 x} \sqrt{3+5 x}}{43904 (2+3 x)}-\frac{2057 \sqrt{1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac{187 \sqrt{1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac{17 \sqrt{1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac{248897 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0619918, size = 84, normalized size = 0.47 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (27422145 x^4+74915550 x^3+74550556 x^2+32206264 x+5112864\right )}{(3 x+2)^5}-3733455 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4609920} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 907229565\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+3024098550\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+4032131400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+383910030\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+2688087600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1048817700\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+896029200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1043707784\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+119470560\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +450887696\,x\sqrt{-10\,{x}^{2}-x+3}+71580096\,\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 = 3.59803, size = 267, normalized size = 1.48 \begin{align*} \frac{248897}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{10285}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{105 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{40 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{6171 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{76109 \, \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 = 1.62103, size = 431, normalized size = 2.39 \begin{align*} -\frac{3733455 \, \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 (27422145 \, x^{4} + 74915550 \, x^{3} + 74550556 \, x^{2} + 32206264 \, x + 5112864\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{9219840 \,{\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 = 3.29242, size = 594, normalized size = 3.3 \begin{align*} \frac{248897}{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{14641 \,{\left (51 \, \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} + 66640 \, \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} + 34119680 \, \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} - 3618944000 \, \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} - 313474560000 \, \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 )}}{65856 \,{\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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