3.2421 \(\int (5-x) \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

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]

(-35*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1152 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/
2))/144 - (2 + 5*x + 3*x^2)^(5/2)/15 + (35*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(2304*Sqrt[3])

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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]

(-35*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1152 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/
2))/144 - (2 + 5*x + 3*x^2)^(5/2)/15 + (35*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 +
5*x + 3*x^2])])/(2304*Sqrt[3])

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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]

35*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/144 - 35*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)
/1152 - (3*x**2 + 5*x + 2)**(5/2)/15 + 35*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqr
t(3*x**2 + 5*x + 2)))/6912

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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]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-11589 - 43070*x - 48792*x^2 - 13680*x^3 + 3456*x^4)
+ 175*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]])/34560

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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]

-1/15*(3*x^2+5*x+2)^(5/2)+35/144*(5+6*x)*(3*x^2+5*x+2)^(3/2)-35/1152*(5+6*x)*(3*
x^2+5*x+2)^(1/2)+35/6912*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

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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]

-1/15*(3*x^2 + 5*x + 2)^(5/2) + 35/24*(3*x^2 + 5*x + 2)^(3/2)*x + 175/144*(3*x^2
+ 5*x + 2)^(3/2) - 35/192*sqrt(3*x^2 + 5*x + 2)*x + 35/6912*sqrt(3)*log(2*sqrt(
3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 175/1152*sqrt(3*x^2 + 5*x + 2)

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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]

-1/69120*sqrt(3)*(4*sqrt(3)*(3456*x^4 - 13680*x^3 - 48792*x^2 - 43070*x - 11589)
*sqrt(3*x^2 + 5*x + 2) - 175*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 +
5*x + 2)*(6*x + 5)))

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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]

-Integral(-23*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-10*sqrt(3*x**
2 + 5*x + 2), x)

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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]

-1/5760*(2*(12*(6*(24*x - 95)*x - 2033)*x - 21535)*x - 11589)*sqrt(3*x^2 + 5*x +
2) - 35/6912*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)
)