Optimal. Leaf size=96 \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.10505, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{7/2}}{42 (3 x+2)^2}-\frac{73 (1-2 x)^{5/2}}{126 (3 x+2)}-\frac{365}{567} (1-2 x)^{3/2}-\frac{365}{81} \sqrt{1-2 x}+\frac{365}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 10.4319, size = 80, normalized size = 0.83 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{42 \left (3 x + 2\right )^{2}} - \frac{73 \left (- 2 x + 1\right )^{\frac{5}{2}}}{126 \left (3 x + 2\right )} - \frac{365 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567} - \frac{365 \sqrt{- 2 x + 1}}{81} + \frac{365 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**3,x)
[Out]
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Mathematica [A] time = 0.104901, size = 63, normalized size = 0.66 \[ \frac{1}{486} \left (\frac{3 \sqrt{1-2 x} \left (720 x^3-4584 x^2-8731 x-3521\right )}{(3 x+2)^2}+730 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x))/(2 + 3*x)^3,x]
[Out]
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Maple [A] time = 0.016, size = 66, normalized size = 0.7 \[ -{\frac{20}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{32}{9}\sqrt{1-2\,x}}-{\frac{28}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{79}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{539}{108}\sqrt{1-2\,x}} \right ) }+{\frac{365\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)/(2+3*x)^3,x)
[Out]
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Maxima [A] time = 1.50192, size = 124, normalized size = 1.29 \[ -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213401, size = 122, normalized size = 1.27 \[ \frac{\sqrt{3}{\left (365 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\right )} \sqrt{-2 \, x + 1}\right )}}{486 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)*(3+5*x)/(2+3*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.212099, size = 116, normalized size = 1.21 \[ -\frac{20}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{365}{486} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{32}{9} \, \sqrt{-2 \, x + 1} + \frac{7 \,{\left (237 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \sqrt{-2 \, x + 1}\right )}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="giac")
[Out]