Optimal. Leaf size=92 \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.178157, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.9975, size = 83, normalized size = 0.9 \[ \frac{\left (- 8856 x + 32760\right ) \sqrt{3 x^{2} + 2}}{1152} + \frac{\left (- 18 x + 156\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{144} - \frac{1529 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{32} - \frac{455 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0923913, size = 126, normalized size = 1.37 \[ \frac{1}{96} \left (312 \sqrt{3 x^2+2} x^2-762 \sqrt{3 x^2+2} x+2938 \sqrt{3 x^2+2}-1365 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-36 \sqrt{3 x^2+2} x^3+1365 \sqrt{35} \log (2 x+3)-4587 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 117, normalized size = 1.3 \[ -{\frac{x}{8} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x}{8}\sqrt{3\,{x}^{2}+2}}-{\frac{1529\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{117\,x}{16}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{455}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{455\,\sqrt{35}}{32}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+2)^(3/2)/(2*x+3),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.787574, size = 126, normalized size = 1.37 \[ -\frac{1}{8} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{13}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{123}{16} \, \sqrt{3 \, x^{2} + 2} x - \frac{1529}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{455}{32} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{455}{16} \, \sqrt{3 \, x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.301139, size = 138, normalized size = 1.5 \[ -\frac{1}{48} \,{\left (18 \, x^{3} - 156 \, x^{2} + 381 \, x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{64} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac{455}{64} \, \sqrt{35} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((5-x)*(3*x**2+2)**(3/2)/(3+2*x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.297023, size = 157, normalized size = 1.71 \[ -\frac{1}{48} \,{\left (3 \,{\left (2 \,{\left (3 \, x - 26\right )} x + 127\right )} x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{32} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{455}{32} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3),x, algorithm="giac")
[Out]