3.1376 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=106 \[ \frac{(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac{3 (111 x+385) \sqrt{3 x^2+2}}{280 (2 x+3)}+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{560 \sqrt{35}}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(-3*(385 + 111*x)*Sqrt[2 + 3*x^2])/(280*(3 + 2*x)) + ((229 + 456*x)*(2 + 3*x^2)^
(3/2))/(420*(3 + 2*x)^3) + (33*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (11727*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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Rubi [A]  time = 0.186128, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(456 x+229) \left (3 x^2+2\right )^{3/2}}{420 (2 x+3)^3}-\frac{3 (111 x+385) \sqrt{3 x^2+2}}{280 (2 x+3)}+\frac{11727 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{560 \sqrt{35}}+\frac{33}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(385 + 111*x)*Sqrt[2 + 3*x^2])/(280*(3 + 2*x)) + ((229 + 456*x)*(2 + 3*x^2)^
(3/2))/(420*(3 + 2*x)^3) + (33*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (11727*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(560*Sqrt[35])

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Rubi in Sympy [A]  time = 18.84, size = 94, normalized size = 0.89 \[ \frac{33 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{16} + \frac{11727 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{19600} - \frac{\left (2664 x + 9240\right ) \sqrt{3 x^{2} + 2}}{2240 \left (2 x + 3\right )} + \frac{\left (912 x + 458\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{840 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33*sqrt(3)*asinh(sqrt(6)*x/2)/16 + 11727*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*
sqrt(3*x**2 + 2)))/19600 - (2664*x + 9240)*sqrt(3*x**2 + 2)/(2240*(2*x + 3)) + (
912*x + 458)*(3*x**2 + 2)**(3/2)/(840*(2*x + 3)**3)

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Mathematica [A]  time = 0.128849, size = 102, normalized size = 0.96 \[ \frac{35181 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{70 \sqrt{3 x^2+2} \left (1260 x^3+24474 x^2+48747 x+30269\right )}{(2 x+3)^3}-35181 \sqrt{35} \log (2 x+3)+121275 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{58800} \]

Antiderivative was successfully verified.

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

[Out]

((-70*Sqrt[2 + 3*x^2]*(30269 + 48747*x + 24474*x^2 + 1260*x^3))/(3 + 2*x)^3 + 12
1275*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 35181*Sqrt[35]*Log[3 + 2*x] + 35181*Sqrt[35]
*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/58800

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Maple [B]  time = 0.016, size = 173, normalized size = 1.6 \[ -{\frac{13}{840} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1}{2450} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{446}{42875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{3909}{85750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{3933\,x}{9800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{33\,\sqrt{3}}{16}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{11727}{19600}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{11727\,\sqrt{35}}{19600}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{1338\,x}{42875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1/2450/(x+3/2)^2*(3*(x+3/2)^2-9*x
-19/4)^(5/2)-446/42875/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)-3909/85750*(3*(x+3/2
)^2-9*x-19/4)^(3/2)+3933/9800*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+33/16*arcsinh(1/2*x
*6^(1/2))*3^(1/2)-11727/19600*(12*(x+3/2)^2-36*x-19)^(1/2)+11727/19600*35^(1/2)*
arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+1338/42875*x*(3*(x+3
/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 0.7852, size = 203, normalized size = 1.92 \[ \frac{3}{2450} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{1225 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{3933}{9800} \, \sqrt{3 \, x^{2} + 2} x + \frac{33}{16} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{11727}{19600} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{11727}{9800} \, \sqrt{3 \, x^{2} + 2} - \frac{223 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2450*(3*x^2 + 2)^(3/2) - 13/105*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27)
- 2/1225*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 3933/9800*sqrt(3*x^2 + 2)*x + 3
3/16*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 11727/19600*sqrt(35)*arcsinh(3/2*sqrt(6)*x
/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 11727/9800*sqrt(3*x^2 + 2) - 223/122
5*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.304701, size = 217, normalized size = 2.05 \[ \frac{\sqrt{35}{\left (3465 \, \sqrt{35} \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - 4 \, \sqrt{35}{\left (1260 \, x^{3} + 24474 \, x^{2} + 48747 \, x + 30269\right )} \sqrt{3 \, x^{2} + 2} + 35181 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} - 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{117600 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/117600*sqrt(35)*(3465*sqrt(35)*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-sqrt(
3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - 4*sqrt(35)*(1260*x^3 + 24474*x^2 + 48747*x +
30269)*sqrt(3*x^2 + 2) + 35181*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(sqrt(35)*(93*
x^2 - 36*x + 43) - 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(8*x^3 + 3
6*x^2 + 54*x + 27)

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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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327899, size = 352, normalized size = 3.32 \[ -\frac{33}{16} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{11727}{19600} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{3}{16} \, \sqrt{3 \, x^{2} + 2} - \frac{44376 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 189285 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 423090 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 561630 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 499440 \, \sqrt{3} x - 50144 \, \sqrt{3} - 499440 \, \sqrt{3 \, x^{2} + 2}}{1120 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/16*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 11727/19600*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))) - 3/16*sqrt(3*x^2 + 2) - 1/1120*(44376*(sqrt(3
)*x - sqrt(3*x^2 + 2))^5 + 189285*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 4230
90*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 561630*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^2 + 499440*sqrt(3)*x - 50144*sqrt(3) - 499440*sqrt(3*x^2 + 2))/((sqrt(3)*x - s
qrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^3