3.1375 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac{(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac{3 (12 x+37) \sqrt{3 x^2+2}}{4 (2 x+3)}-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8 \sqrt{35}}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(37 + 12*x)*Sqrt[2 + 3*x^2])/(4*(3 + 2*x)) - ((8 + x)*(2 + 3*x^2)^(3/2))/(4*(
3 + 2*x)^2) - (111*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (1143*ArcTanh[(4 - 9*x)/(Sq
rt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

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Rubi [A]  time = 0.182202, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(x+8) \left (3 x^2+2\right )^{3/2}}{4 (2 x+3)^2}+\frac{3 (12 x+37) \sqrt{3 x^2+2}}{4 (2 x+3)}-\frac{1143 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8 \sqrt{35}}-\frac{111}{8} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(37 + 12*x)*Sqrt[2 + 3*x^2])/(4*(3 + 2*x)) - ((8 + x)*(2 + 3*x^2)^(3/2))/(4*(
3 + 2*x)^2) - (111*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/8 - (1143*ArcTanh[(4 - 9*x)/(Sq
rt[35]*Sqrt[2 + 3*x^2])])/(8*Sqrt[35])

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Rubi in Sympy [A]  time = 18.8364, size = 95, normalized size = 0.91 \[ - \frac{111 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{8} - \frac{1143 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{280} + \frac{3 \left (384 x + 1184\right ) \sqrt{3 x^{2} + 2}}{128 \left (2 x + 3\right )} - \frac{\left (4 x + 32\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{16 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-111*sqrt(3)*asinh(sqrt(6)*x/2)/8 - 1143*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*
sqrt(3*x**2 + 2)))/280 + 3*(384*x + 1184)*sqrt(3*x**2 + 2)/(128*(2*x + 3)) - (4*
x + 32)*(3*x**2 + 2)**(3/2)/(16*(2*x + 3)**2)

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Mathematica [A]  time = 0.147358, size = 102, normalized size = 0.98 \[ \frac{1}{280} \left (-1143 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{70 \sqrt{3 x^2+2} \left (3 x^3-48 x^2-328 x-317\right )}{(2 x+3)^2}+1143 \sqrt{35} \log (2 x+3)-3885 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-70*Sqrt[2 + 3*x^2]*(-317 - 328*x - 48*x^2 + 3*x^3))/(3 + 2*x)^2 - 3885*Sqrt[3
]*ArcSinh[Sqrt[3/2]*x] + 1143*Sqrt[35]*Log[3 + 2*x] - 1143*Sqrt[35]*Log[2*(4 - 9
*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/280

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Maple [A]  time = 0.015, size = 152, normalized size = 1.5 \[ -{\frac{13}{280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{187}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{381}{1225} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{171\,x}{70}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{111\,\sqrt{3}}{8}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{1143}{280}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{1143\,\sqrt{35}}{280}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }-{\frac{561\,x}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/280/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)+187/4900/(x+3/2)*(3*(x+3/2)^2-9*x
-19/4)^(5/2)+381/1225*(3*(x+3/2)^2-9*x-19/4)^(3/2)-171/70*x*(3*(x+3/2)^2-9*x-19/
4)^(1/2)-111/8*arcsinh(1/2*x*6^(1/2))*3^(1/2)+1143/280*(12*(x+3/2)^2-36*x-19)^(1
/2)-1143/280*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2)
)-561/4900*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 0.782418, size = 165, normalized size = 1.59 \[ \frac{39}{280} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{171}{70} \, \sqrt{3 \, x^{2} + 2} x - \frac{111}{8} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{1143}{280} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{1143}{140} \, \sqrt{3 \, x^{2} + 2} + \frac{187 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{280 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39/280*(3*x^2 + 2)^(3/2) - 13/70*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 171/70*s
qrt(3*x^2 + 2)*x - 111/8*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1143/280*sqrt(35)*arcs
inh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1143/140*sqrt(3*x^2
+ 2) + 187/280*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.302575, size = 196, normalized size = 1.88 \[ \frac{\sqrt{35}{\left (111 \, \sqrt{35} \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - 4 \, \sqrt{35}{\left (3 \, x^{3} - 48 \, x^{2} - 328 \, x - 317\right )} \sqrt{3 \, x^{2} + 2} + 1143 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{560 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/560*sqrt(35)*(111*sqrt(35)*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 +
2)*x - 3*x^2 - 1) - 4*sqrt(35)*(3*x^3 - 48*x^2 - 328*x - 317)*sqrt(3*x^2 + 2) +
1143*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2
)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(4*x^2 + 12*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.300916, size = 296, normalized size = 2.85 \[ -\frac{3}{16} \, \sqrt{3 \, x^{2} + 2}{\left (x - 19\right )} + \frac{111}{8} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{1143}{280} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{5 \,{\left (1452 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 3013 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 6528 \, \sqrt{3} x + 1048 \, \sqrt{3} + 6528 \, \sqrt{3 \, x^{2} + 2}\right )}}{64 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/16*sqrt(3*x^2 + 2)*(x - 19) + 111/8*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))
+ 1143/280*sqrt(35)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 +
2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 5/64*(1452*(sqr
t(3)*x - sqrt(3*x^2 + 2))^3 + 3013*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 652
8*sqrt(3)*x + 1048*sqrt(3) + 6528*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2)
)^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2