3.1405 \(\int \frac{5-x}{(3+2 x)^4 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{10 \sqrt{3 x^2+2}}{343 (2 x+3)}-\frac{16 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+2}}{105 (2 x+3)^3}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) - (16*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2)
- (10*Sqrt[2 + 3*x^2])/(343*(3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/(1715*Sqrt[35])

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Rubi [A]  time = 0.182375, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{10 \sqrt{3 x^2+2}}{343 (2 x+3)}-\frac{16 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{13 \sqrt{3 x^2+2}}{105 (2 x+3)^3}-\frac{57 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1715 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) - (16*Sqrt[2 + 3*x^2])/(245*(3 + 2*x)^2)
- (10*Sqrt[2 + 3*x^2])/(343*(3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/(1715*Sqrt[35])

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Rubi in Sympy [A]  time = 19.2851, size = 90, normalized size = 0.91 \[ - \frac{57 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{60025} - \frac{10 \sqrt{3 x^{2} + 2}}{343 \left (2 x + 3\right )} - \frac{16 \sqrt{3 x^{2} + 2}}{245 \left (2 x + 3\right )^{2}} - \frac{13 \sqrt{3 x^{2} + 2}}{105 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**4/(3*x**2+2)**(1/2),x)

[Out]

-57*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/60025 - 10*sqrt(3*
x**2 + 2)/(343*(2*x + 3)) - 16*sqrt(3*x**2 + 2)/(245*(2*x + 3)**2) - 13*sqrt(3*x
**2 + 2)/(105*(2*x + 3)**3)

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Mathematica [A]  time = 0.128027, size = 80, normalized size = 0.81 \[ \frac{-\frac{35 \sqrt{3 x^2+2} \left (600 x^2+2472 x+2995\right )}{(2 x+3)^3}-171 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+171 \sqrt{35} \log (2 x+3)}{180075} \]

Antiderivative was successfully verified.

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(2995 + 2472*x + 600*x^2))/(3 + 2*x)^3 + 171*Sqrt[35]*Log[
3 + 2*x] - 171*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/180075

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Maple [A]  time = 0.016, size = 95, normalized size = 1. \[ -{\frac{13}{840}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{4}{245}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{5}{343}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{57\,\sqrt{35}}{60025}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(2*x+3)^4/(3*x^2+2)^(1/2),x)

[Out]

-13/840/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(1/2)-4/245/(x+3/2)^2*(3*(x+3/2)^2-9*x-
19/4)^(1/2)-5/343/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)-57/60025*35^(1/2)*arctanh
(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 0.755785, size = 140, normalized size = 1.41 \[ \frac{57}{60025} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{105 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{16 \, \sqrt{3 \, x^{2} + 2}}{245 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{10 \, \sqrt{3 \, x^{2} + 2}}{343 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

57/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3))
- 13/105*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 16/245*sqrt(3*x^2 + 2)/
(4*x^2 + 12*x + 9) - 10/343*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 0.277967, size = 149, normalized size = 1.51 \[ -\frac{\sqrt{35}{\left (2 \, \sqrt{35}{\left (600 \, x^{2} + 2472 \, x + 2995\right )} \sqrt{3 \, x^{2} + 2} - 171 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{360150 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/360150*sqrt(35)*(2*sqrt(35)*(600*x^2 + 2472*x + 2995)*sqrt(3*x^2 + 2) - 171*(
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 + 36*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+2*x)**4/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.306664, size = 308, normalized size = 3.11 \[ \frac{57}{60025} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{114 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 855 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 6750 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 13290 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 10344 \, \sqrt{3} x - 800 \, \sqrt{3} - 10344 \, \sqrt{3 \, x^{2} + 2}}{3430 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

57/60025*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))) - 1/3430*(114*(sqrt
(3)*x - sqrt(3*x^2 + 2))^5 + 855*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 6750*
(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 13290*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2
+ 10344*sqrt(3)*x - 800*sqrt(3) - 10344*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)^3