Optimal. Leaf size=118 \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]
[Out]
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Rubi [A] time = 0.203561, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 20.1884, size = 97, normalized size = 0.82 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (192375 x + 56925\right )}{17010} - \frac{6 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{6 \left (3 x + 2\right )^{2}} - \frac{887 \sqrt{- 2 x + 1}}{189} + \frac{887 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{567} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**3,x)
[Out]
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Mathematica [A] time = 0.107982, size = 63, normalized size = 0.53 \[ \frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (1800 x^4+570 x^2+2965 x+1367\right )}{54 (3 x+2)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^3,x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.6 \[ -{\frac{25}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{370}{81}\sqrt{1-2\,x}}-{\frac{2}{9\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{215}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{497}{18}\sqrt{1-2\,x}} \right ) }+{\frac{887\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^3,x)
[Out]
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Maxima [A] time = 1.49383, size = 136, normalized size = 1.15 \[ -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*(-2*x + 1)^(3/2)/(3*x + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211789, size = 113, normalized size = 0.96 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1800 \, x^{4} + 570 \, x^{2} + 2965 \, x + 1367\right )} \sqrt{-2 \, x + 1} - 887 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1134 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^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)**3/(2+3*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.223522, size = 138, normalized size = 1.17 \[ -\frac{25}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*(-2*x + 1)^(3/2)/(3*x + 2)^3,x, algorithm="giac")
[Out]