Optimal. Leaf size=151 \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac{1991 (5 x+3)^{3/2} \sqrt{1-2 x}}{224 (3 x+2)^2}-\frac{21901 \sqrt{5 x+3} \sqrt{1-2 x}}{3136 (3 x+2)}-\frac{240911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]
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Rubi [A] time = 0.0417163, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac{1991 (5 x+3)^{3/2} \sqrt{1-2 x}}{224 (3 x+2)^2}-\frac{21901 \sqrt{5 x+3} \sqrt{1-2 x}}{3136 (3 x+2)}-\frac{240911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^5} \, dx &=\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181}{56} \int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac{1991}{112} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac{1991 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{21901}{448} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}+\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac{1991 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{240911 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6272}\\ &=-\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}+\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac{1991 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{240911 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{3136}\\ &=-\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}+\frac{3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac{1991 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac{240911 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{3136 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0540639, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (1705089 x^3+3485960 x^2+2381420 x+541680\right )}{(3 x+2)^4}-722733 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{65856} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{131712\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 58541373\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+156110328\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+156110328\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+23871246\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+69382368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+48803440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11563728\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +33339880\,x\sqrt{-10\,{x}^{2}-x+3}+7583520\,\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 = 3.65803, size = 212, normalized size = 1.4 \begin{align*} \frac{240911}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{9955}{2352} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{168 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{5973 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1568 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{73667 \, \sqrt{-10 \, x^{2} - x + 3}}{9408 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51686, size = 367, normalized size = 2.43 \begin{align*} -\frac{722733 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (1705089 \, x^{3} + 3485960 \, x^{2} + 2381420 \, x + 541680\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{131712 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 3.01846, size = 512, normalized size = 3.39 \begin{align*} \frac{240911}{439040} \, \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 (543 \, \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} - 696920 \, \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} - 156094400 \, \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} - 11919936000 \, \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 )}}{4704 \,{\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 )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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