3.1408 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{7 (2-7 x) (2 x+3)^3}{6 \sqrt{3 x^2+2}}-\frac{151}{27} \sqrt{3 x^2+2} (2 x+3)^2-\frac{10}{81} (207 x+185) \sqrt{3 x^2+2}+\frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(6*Sqrt[2 + 3*x^2]) - (151*(3 + 2*x)^2*Sqrt[2 + 3*x^2
])/27 - (10*(185 + 207*x)*Sqrt[2 + 3*x^2])/81 + (880*ArcSinh[Sqrt[3/2]*x])/(3*Sq
rt[3])

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Rubi [A]  time = 0.158604, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{7 (2-7 x) (2 x+3)^3}{6 \sqrt{3 x^2+2}}-\frac{151}{27} \sqrt{3 x^2+2} (2 x+3)^2-\frac{10}{81} (207 x+185) \sqrt{3 x^2+2}+\frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(3/2),x]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(6*Sqrt[2 + 3*x^2]) - (151*(3 + 2*x)^2*Sqrt[2 + 3*x^2
])/27 - (10*(185 + 207*x)*Sqrt[2 + 3*x^2])/81 + (880*ArcSinh[Sqrt[3/2]*x])/(3*Sq
rt[3])

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Rubi in Sympy [A]  time = 14.7963, size = 78, normalized size = 0.88 \[ - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{3}}{12 \sqrt{3 x^{2} + 2}} - \frac{151 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 2}}{27} - \frac{\left (16560 x + 14800\right ) \sqrt{3 x^{2} + 2}}{648} + \frac{880 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4/(3*x**2+2)**(3/2),x)

[Out]

-(-98*x + 28)*(2*x + 3)**3/(12*sqrt(3*x**2 + 2)) - 151*(2*x + 3)**2*sqrt(3*x**2
+ 2)/27 - (16560*x + 14800)*sqrt(3*x**2 + 2)/648 + 880*sqrt(3)*asinh(sqrt(6)*x/2
)/9

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Mathematica [A]  time = 0.0971718, size = 55, normalized size = 0.62 \[ \frac{880 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}-\frac{288 x^4+432 x^3-15024 x^2+14715 x+33914}{162 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(3/2),x]

[Out]

-(33914 + 14715*x - 15024*x^2 + 432*x^3 + 288*x^4)/(162*Sqrt[2 + 3*x^2]) + (880*
ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Maple [A]  time = 0.02, size = 79, normalized size = 0.9 \[ -{\frac{545\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{16957}{81}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{880\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{2504\,{x}^{2}}{27}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{8\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{16\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(2*x+3)^4/(3*x^2+2)^(3/2),x)

[Out]

-545/6*x/(3*x^2+2)^(1/2)-16957/81/(3*x^2+2)^(1/2)+880/9*arcsinh(1/2*x*6^(1/2))*3
^(1/2)+2504/27*x^2/(3*x^2+2)^(1/2)-8/3*x^3/(3*x^2+2)^(1/2)-16/9*x^4/(3*x^2+2)^(1
/2)

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Maxima [A]  time = 0.756126, size = 105, normalized size = 1.18 \[ -\frac{16 \, x^{4}}{9 \, \sqrt{3 \, x^{2} + 2}} - \frac{8 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{2504 \, x^{2}}{27 \, \sqrt{3 \, x^{2} + 2}} + \frac{880}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{545 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{16957}{81 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="maxima")

[Out]

-16/9*x^4/sqrt(3*x^2 + 2) - 8/3*x^3/sqrt(3*x^2 + 2) + 2504/27*x^2/sqrt(3*x^2 + 2
) + 880/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 545/6*x/sqrt(3*x^2 + 2) - 16957/81/sq
rt(3*x^2 + 2)

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Fricas [A]  time = 0.273866, size = 112, normalized size = 1.26 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (288 \, x^{4} + 432 \, x^{3} - 15024 \, x^{2} + 14715 \, x + 33914\right )} \sqrt{3 \, x^{2} + 2} - 23760 \,{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{486 \,{\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="fricas")

[Out]

-1/486*sqrt(3)*(sqrt(3)*(288*x^4 + 432*x^3 - 15024*x^2 + 14715*x + 33914)*sqrt(3
*x^2 + 2) - 23760*(3*x^2 + 2)*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))/(
3*x^2 + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{999 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \left (- \frac{864 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \left (- \frac{264 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \frac{16 x^{4}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \frac{16 x^{5}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \left (- \frac{405}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4/(3*x**2+2)**(3/2),x)

[Out]

-Integral(-999*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-
864*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-264*x**3
/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(16*x**4/(3*x**2*s
qrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(16*x**5/(3*x**2*sqrt(3*x**2
+ 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-405/(3*x**2*sqrt(3*x**2 + 2) + 2*sqr
t(3*x**2 + 2)), x)

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GIAC/XCAS [A]  time = 0.328779, size = 73, normalized size = 0.82 \[ -\frac{880}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (16 \,{\left (3 \,{\left (2 \, x + 3\right )} x - 313\right )} x + 4905\right )} x + 33914}{162 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="giac")

[Out]

-880/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(16*(3*(2*x + 3)*x -
313)*x + 4905)*x + 33914)/sqrt(3*x^2 + 2)