Optimal. Leaf size=144 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0538979, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}-\frac{1}{18} \int \frac{\left (-\frac{915}{4}-285 x\right ) \sqrt{1-2 x}}{(2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}-\frac{1}{270} \int \frac{-\frac{14865}{4}-3540 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{118}{27} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{1085}{216} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{1085}{108} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{236 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{27 \sqrt{5}}\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.145071, size = 126, normalized size = 0.88 \[ \frac{-15 \sqrt{5 x+3} \left (96 x^3+822 x^2+37 x-236\right )-472 \sqrt{10-20 x} (3 x+2)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-775 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{540 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 208, normalized size = 1.4 \begin{align*}{\frac{1}{1080\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6975\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4248\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+9300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5664\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1888\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +13050\,x\sqrt{-10\,{x}^{2}-x+3}+7080\,\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 = 5.78158, size = 136, normalized size = 0.94 \begin{align*} \frac{59}{135} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{155}{216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{13}{9} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59275, size = 437, normalized size = 3.03 \begin{align*} -\frac{472 \, \sqrt{5} \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 775 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\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 ) - 30 \,{\left (48 \, x^{2} + 435 \, x + 236\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1080 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 2.31783, size = 463, normalized size = 3.22 \begin{align*} \frac{31}{432} \, \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{59}{135} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{135} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{77 \,{\left (17 \, \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} - 13720 \, \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 )}}{54 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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