Optimal. Leaf size=81 \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.0737455, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{5/2}}{42 (3 x+2)^2}-\frac{71 (1-2 x)^{3/2}}{126 (3 x+2)}-\frac{71}{63} \sqrt{1-2 x}+\frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 9.07737, size = 68, normalized size = 0.84 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{42 \left (3 x + 2\right )^{2}} - \frac{71 \left (- 2 x + 1\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )} - \frac{71 \sqrt{- 2 x + 1}}{63} + \frac{71 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{189} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(3+5*x)/(2+3*x)**3,x)
[Out]
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Mathematica [A] time = 0.100154, size = 58, normalized size = 0.72 \[ \frac{71 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (120 x^2+235 x+101\right )}{18 (3 x+2)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(3 + 5*x))/(2 + 3*x)^3,x]
[Out]
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Maple [A] time = 0.016, size = 57, normalized size = 0.7 \[ -{\frac{20}{27}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{25}{4} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{71\,\sqrt{21}}{189}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(3+5*x)/(2+3*x)^3,x)
[Out]
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Maxima [A] time = 1.53401, size = 112, normalized size = 1.38 \[ -\frac{71}{378} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{27 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236758, size = 107, normalized size = 1.32 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (120 \, x^{2} + 235 \, x + 101\right )} \sqrt{-2 \, x + 1} - 71 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{378 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(3/2)/(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)**(3/2)*(3+5*x)/(2+3*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.242581, size = 104, normalized size = 1.28 \[ -\frac{71}{378} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} + \frac{225 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 511 \, \sqrt{-2 \, x + 1}}{108 \,{\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^3,x, algorithm="giac")
[Out]