Optimal. Leaf size=142 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{12 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{205}{36} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{37}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1649 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]
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Rubi [A] time = 0.0543256, antiderivative size = 142, 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} \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{12 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{205}{36} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{37}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1649 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]
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)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{12 (2+3 x)}-\frac{1}{18} \int \frac{\left (\frac{9}{4}-615 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{205}{36} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{12 (2+3 x)}+\frac{1}{108} \int \frac{-\frac{1311}{2}-2220 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{205}{36} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{12 (2+3 x)}-\frac{185}{27} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{1649}{216} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{205}{36} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{12 (2+3 x)}+\frac{1649}{108} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{1}{27} \left (74 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{205}{36} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{12 (2+3 x)}-\frac{37}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{1649 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{108 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.133466, size = 126, normalized size = 0.89 \[ \frac{21 \sqrt{5 x+3} \left (240 x^3+570 x^2-x-172\right )+1036 \sqrt{10-20 x} (3 x+2)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-1649 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{756 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 208, normalized size = 1.5 \begin{align*}{\frac{1}{1512\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 14841\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-9324\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+19788\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-12432\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-5040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6596\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4144\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -14490\,x\sqrt{-10\,{x}^{2}-x+3}-7224\,\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 = 1.81091, size = 176, normalized size = 1.24 \begin{align*} \frac{5}{21} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{185}{42} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{37}{54} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1649}{1512} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{769}{252} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{84 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55461, size = 423, normalized size = 2.98 \begin{align*} -\frac{1649 \, \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 ) - 1036 \, \sqrt{10}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \,{\left (120 \, x^{2} + 345 \, x + 172\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1512 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (1 - 2 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\left (3 x + 2\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.78469, size = 463, normalized size = 3.26 \begin{align*} \frac{1649}{15120} \, \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{37}{54} \, \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{2}{27} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{55 \,{\left (23 \, \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} + 10136 \, \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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