3.1413 \(\int \frac{5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{41 x+26}{70 \sqrt{3 x^2+2}}-\frac{26 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) - (26*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])])/(35*Sqrt[35])

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Rubi [A]  time = 0.0813166, antiderivative size = 53, 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 \[ \frac{41 x+26}{70 \sqrt{3 x^2+2}}-\frac{26 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) - (26*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^
2])])/(35*Sqrt[35])

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Rubi in Sympy [A]  time = 10.942, size = 48, normalized size = 0.91 \[ \frac{123 x + 78}{210 \sqrt{3 x^{2} + 2}} - \frac{26 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1225} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(123*x + 78)/(210*sqrt(3*x**2 + 2)) - 26*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*
sqrt(3*x**2 + 2)))/1225

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Mathematica [A]  time = 0.102282, size = 78, normalized size = 1.47 \[ \frac{\frac{-52 \sqrt{35} \sqrt{3 x^2+2} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+1435 x+910}{\sqrt{3 x^2+2}}+52 \sqrt{35} \log (2 x+3)}{2450} \]

Antiderivative was successfully verified.

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

[Out]

(52*Sqrt[35]*Log[3 + 2*x] + (910 + 1435*x - 52*Sqrt[35]*Sqrt[2 + 3*x^2]*Log[2*(4
- 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/Sqrt[2 + 3*x^2])/2450

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Maple [A]  time = 0.011, size = 77, normalized size = 1.5 \[ -{\frac{x}{4}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{13}{35}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{117\,x}{140}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{26\,\sqrt{35}}{1225}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x/(3*x^2+2)^(1/2)+13/35/(3*(x+3/2)^2-9*x-19/4)^(1/2)+117/140*x/(3*(x+3/2)^2
-9*x-19/4)^(1/2)-26/1225*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 = 0.768616, size = 78, normalized size = 1.47 \[ \frac{26}{1225} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{41 \, x}{70 \, \sqrt{3 \, x^{2} + 2}} + \frac{13}{35 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

26/1225*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3))
+ 41/70*x/sqrt(3*x^2 + 2) + 13/35/sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.275144, size = 119, normalized size = 2.25 \[ \frac{\sqrt{35}{\left (\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (41 \, x + 26\right )} + 26 \,{\left (3 \, x^{2} + 2\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{2450 \,{\left (3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2450*sqrt(35)*(sqrt(35)*sqrt(3*x^2 + 2)*(41*x + 26) + 26*(3*x^2 + 2)*log(-(sqr
t(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/
(3*x^2 + 2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.346813, size = 113, normalized size = 2.13 \[ \frac{26}{1225} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{41 \, x + 26}{70 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

26/1225*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))) + 1/70*(41*x + 26)/s
qrt(3*x^2 + 2)