Optimal. Leaf size=166 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{35}{4} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{185}{27} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1945}{324} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{6829 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{324 \sqrt{7}} \]
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Rubi [A] time = 0.063557, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{35}{4} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{185}{27} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1945}{324} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{6829 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{324 \sqrt{7}} \]
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)^{5/2}}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac{1}{18} \int \frac{\left (\frac{1789}{4}-1890 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{35}{4} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac{1}{216} \int \frac{(909-8880 x) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{185}{27} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{35}{4} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac{\int \frac{-25242-58350 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1296}\\ &=\frac{185}{27} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{35}{4} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac{6829}{648} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{9725}{648} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{185}{27} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{35}{4} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac{6829}{324} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{1}{324} \left (1945 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=\frac{185}{27} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{35}{4} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac{1945}{324} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{6829 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{324 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.157103, size = 131, normalized size = 0.79 \[ \frac{42 \sqrt{5 x+3} \left (1800 x^4-3090 x^3-4875 x^2+521 x+1232\right )-13615 \sqrt{10-20 x} (3 x+2)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+13658 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 225, normalized size = 1.4 \begin{align*}{\frac{1}{9072\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 122535\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-122922\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-75600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+163380\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-163896\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+91980\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+54460\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -54632\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +250740\,x\sqrt{-10\,{x}^{2}-x+3}+103488\,\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 = 2.80597, size = 176, normalized size = 1.06 \begin{align*} -\frac{5}{63} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{535}{126} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1945}{1296} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{6829}{4536} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1627}{378} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{59 \,{\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 = 1.59762, size = 462, normalized size = 2.78 \begin{align*} -\frac{13615 \, \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 ) - 13658 \, \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 ) + 84 \,{\left (900 \, x^{3} - 1095 \, x^{2} - 2985 \, x - 1232\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{9072 \,{\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 = 3.15112, size = 481, normalized size = 2.9 \begin{align*} -\frac{6829}{45360} \, \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{1}{108} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 63 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1945}{1296} \, \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{55 \,{\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} + 5992 \, \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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