Optimal. Leaf size=103 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac{35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac{35 (6 x+5) \sqrt{3 x^2+5 x+2}}{1152}+\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{2304 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0709044, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac{35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac{35 (6 x+5) \sqrt{3 x^2+5 x+2}}{1152}+\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{2304 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(5 - x)*(2 + 5*x + 3*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.91495, size = 94, normalized size = 0.91 \[ \frac{35 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{144} - \frac{35 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{1152} - \frac{\left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{15} + \frac{35 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{6912} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+5*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0579358, size = 70, normalized size = 0.68 \[ \frac{175 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )-6 \sqrt{3 x^2+5 x+2} \left (3456 x^4-13680 x^3-48792 x^2-43070 x-11589\right )}{34560} \]
Antiderivative was successfully verified.
[In] Integrate[(5 - x)*(2 + 5*x + 3*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 83, normalized size = 0.8 \[ -{\frac{1}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{175+210\,x}{144} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{175+210\,x}{1152}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{35\,\sqrt{3}}{6912}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+5*x+2)^(3/2),x)
[Out]
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Maxima [A] time = 0.771729, size = 136, normalized size = 1.32 \[ -\frac{1}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{35}{24} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{175}{144} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{35}{192} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{35}{6912} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{175}{1152} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279244, size = 108, normalized size = 1.05 \[ -\frac{1}{69120} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (3456 \, x^{4} - 13680 \, x^{3} - 48792 \, x^{2} - 43070 \, x - 11589\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 175 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- 23 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 3 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 10 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3*x**2+5*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274396, size = 93, normalized size = 0.9 \[ -\frac{1}{5760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (24 \, x - 95\right )} x - 2033\right )} x - 21535\right )} x - 11589\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{35}{6912} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(3/2)*(x - 5),x, algorithm="giac")
[Out]