3.1404 \(\int \frac{5-x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{281 \sqrt{3 x^2+2}}{2450 (2 x+3)}-\frac{13 \sqrt{3 x^2+2}}{70 (2 x+3)^2}-\frac{291 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(70*(3 + 2*x)^2) - (281*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x))
- (291*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

_______________________________________________________________________________________

Rubi [A]  time = 0.127048, antiderivative size = 77, 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 \[ -\frac{281 \sqrt{3 x^2+2}}{2450 (2 x+3)}-\frac{13 \sqrt{3 x^2+2}}{70 (2 x+3)^2}-\frac{291 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(70*(3 + 2*x)^2) - (281*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x))
- (291*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3463, size = 70, normalized size = 0.91 \[ - \frac{291 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{42875} - \frac{281 \sqrt{3 x^{2} + 2}}{2450 \left (2 x + 3\right )} - \frac{13 \sqrt{3 x^{2} + 2}}{70 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-291*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/42875 - 281*sqrt(
3*x**2 + 2)/(2450*(2*x + 3)) - 13*sqrt(3*x**2 + 2)/(70*(2*x + 3)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.121847, size = 75, normalized size = 0.97 \[ \frac{-\frac{35 \sqrt{3 x^2+2} (281 x+649)}{(2 x+3)^2}-291 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+291 \sqrt{35} \log (2 x+3)}{42875} \]

Antiderivative was successfully verified.

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

[Out]

((-35*(649 + 281*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 + 291*Sqrt[35]*Log[3 + 2*x] - 2
91*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/42875

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 74, normalized size = 1. \[ -{\frac{13}{280}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{281}{4900}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{291\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/280/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-281/4900/(x+3/2)*(3*(x+3/2)^2-9*x
-19/4)^(1/2)-291/42875*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x
-19)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.761931, size = 103, normalized size = 1.34 \[ \frac{291}{42875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{281 \, \sqrt{3 \, x^{2} + 2}}{2450 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

291/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)
) - 13/70*sqrt(3*x^2 + 2)/(4*x^2 + 12*x + 9) - 281/2450*sqrt(3*x^2 + 2)/(2*x + 3
)

_______________________________________________________________________________________

Fricas [A]  time = 0.27629, size = 128, normalized size = 1.66 \[ -\frac{\sqrt{35}{\left (2 \, \sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (281 \, x + 649\right )} - 291 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{85750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/85750*sqrt(35)*(2*sqrt(35)*sqrt(3*x^2 + 2)*(281*x + 649) - 291*(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)

_______________________________________________________________________________________

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)**3/(3*x**2+2)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.303365, size = 247, normalized size = 3.21 \[ \frac{291}{42875} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{1164 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 6463 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 17904 \, \sqrt{3} x + 2248 \, \sqrt{3} + 17904 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

291/42875*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))) - 1/4900*(1164*(sq
rt(3)*x - sqrt(3*x^2 + 2))^3 + 6463*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 17
904*sqrt(3)*x + 2248*sqrt(3) + 17904*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