Optimal. Leaf size=100 \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.158039, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(5 - x)*(3 + 2*x)^3*Sqrt[2 + 3*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.2638, size = 88, normalized size = 0.88 \[ \frac{511 x \sqrt{3 x^{2} + 2}}{9} - \frac{\left (2 x + 3\right )^{3} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{18} + \frac{17 \left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{30} + \frac{\left (67284 x + 226296\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9720} + \frac{1022 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3+2*x)**3*(3*x**2+2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0595197, size = 60, normalized size = 0.6 \[ \frac{1}{270} \left (10220 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (360 x^5-216 x^4-8445 x^3-21918 x^2-21120 x-14516\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(5 - x)*(3 + 2*x)^3*Sqrt[2 + 3*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 77, normalized size = 0.8 \[{\frac{511\,x}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{1022\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{3629}{135} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{193\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(2*x+3)^3*(3*x^2+2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.775498, size = 103, normalized size = 1.03 \[ -\frac{4}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + \frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{193}{18} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{3629}{135} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{511}{9} \, \sqrt{3 \, x^{2} + 2} x + \frac{1022}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^3*(x - 5),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.286617, size = 97, normalized size = 0.97 \[ -\frac{1}{810} \, \sqrt{3}{\left (\sqrt{3}{\left (360 \, x^{5} - 216 \, x^{4} - 8445 \, x^{3} - 21918 \, x^{2} - 21120 \, x - 14516\right )} \sqrt{3 \, x^{2} + 2} - 15330 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^3*(x - 5),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 53.8152, size = 150, normalized size = 1.5 \[ - \frac{4 x^{7}}{\sqrt{3 x^{2} + 2}} + \frac{547 x^{5}}{6 \sqrt{3 x^{2} + 2}} + \frac{1705 x^{3}}{18 \sqrt{3 x^{2} + 2}} + \frac{135 x \sqrt{3 x^{2} + 2}}{2} + \frac{193 x}{9 \sqrt{3 x^{2} + 2}} + \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{5}{2}}}{45} - \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{2}}}{27} + 27 \left (3 x^{2} + 2\right )^{\frac{3}{2}} + \frac{1022 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3+2*x)**3*(3*x**2+2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.291213, size = 77, normalized size = 0.77 \[ -\frac{1}{270} \,{\left (3 \,{\left ({\left ({\left (24 \,{\left (5 \, x - 3\right )} x - 2815\right )} x - 7306\right )} x - 7040\right )} x - 14516\right )} \sqrt{3 \, x^{2} + 2} - \frac{1022}{27} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 2)*(2*x + 3)^3*(x - 5),x, algorithm="giac")
[Out]