3.1409 \(\int \frac{(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{7 (2-7 x) (2 x+3)^2}{6 \sqrt{3 x^2+2}}-\frac{2}{9} (51 x+131) \sqrt{3 x^2+2}+\frac{134 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) - (2*(131 + 51*x)*Sqrt[2 + 3*x^2]
)/9 + (134*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.105292, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{7 (2-7 x) (2 x+3)^2}{6 \sqrt{3 x^2+2}}-\frac{2}{9} (51 x+131) \sqrt{3 x^2+2}+\frac{134 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) - (2*(131 + 51*x)*Sqrt[2 + 3*x^2]
)/9 + (134*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 10.3525, size = 58, normalized size = 0.87 \[ - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{2}}{12 \sqrt{3 x^{2} + 2}} - \frac{\left (816 x + 2096\right ) \sqrt{3 x^{2} + 2}}{72} + \frac{134 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-98*x + 28)*(2*x + 3)**2/(12*sqrt(3*x**2 + 2)) - (816*x + 2096)*sqrt(3*x**2 +
2)/72 + 134*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Mathematica [A]  time = 0.0639124, size = 63, normalized size = 0.94 \[ -\frac{\sqrt{3 x^2+2} \left (24 x^3-24 x^2-411 x+1426\right )-268 \sqrt{3} \left (3 x^2+2\right ) \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{54 x^2+36} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2 + 3*x^2]*(1426 - 411*x - 24*x^2 + 24*x^3) - 268*Sqrt[3]*(2 + 3*x^2)*Ar
cSinh[Sqrt[3/2]*x])/(36 + 54*x^2))

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Maple [A]  time = 0.009, size = 65, normalized size = 1. \[{\frac{137\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{713}{9}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{134\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{4\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

137/6*x/(3*x^2+2)^(1/2)-713/9/(3*x^2+2)^(1/2)+134/9*arcsinh(1/2*x*6^(1/2))*3^(1/
2)+4/3*x^2/(3*x^2+2)^(1/2)-4/3*x^3/(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.755162, size = 86, normalized size = 1.28 \[ -\frac{4 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{4 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 2}} + \frac{134}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{137 \, x}{6 \, \sqrt{3 \, x^{2} + 2}} - \frac{713}{9 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3*x^3/sqrt(3*x^2 + 2) + 4/3*x^2/sqrt(3*x^2 + 2) + 134/9*sqrt(3)*arcsinh(1/2*s
qrt(6)*x) + 137/6*x/sqrt(3*x^2 + 2) - 713/9/sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.276851, size = 105, normalized size = 1.57 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (24 \, x^{3} - 24 \, x^{2} - 411 \, x + 1426\right )} \sqrt{3 \, x^{2} + 2} - 402 \,{\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{54 \,{\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(24*x^3 - 24*x^2 - 411*x + 1426)*sqrt(3*x^2 + 2) - 402*(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. \[ - \int \left (- \frac{243 x}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \left (- \frac{126 x^{2}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \left (- \frac{4 x^{3}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx - \int \frac{8 x^{4}}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\, dx - \int \left (- \frac{135}{3 x^{2} \sqrt{3 x^{2} + 2} + 2 \sqrt{3 x^{2} + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-243*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-
126*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-4*x**3/(
3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(8*x**4/(3*x**2*sqrt
(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-135/(3*x**2*sqrt(3*x**2 + 2)
+ 2*sqrt(3*x**2 + 2)), x)

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GIAC/XCAS [A]  time = 0.28403, size = 63, normalized size = 0.94 \[ -\frac{134}{9} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left (8 \,{\left (x - 1\right )} x - 137\right )} x + 1426}{18 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-134/9*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*(x - 1)*x - 137)*x
+ 1426)/sqrt(3*x^2 + 2)