3.1403 \(\int \frac{5-x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{13 \sqrt{3 x^2+2}}{35 (2 x+3)}-\frac{41 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (41*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 +
3*x^2])])/(35*Sqrt[35])

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Rubi [A]  time = 0.0764289, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{13 \sqrt{3 x^2+2}}{35 (2 x+3)}-\frac{41 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (41*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 +
3*x^2])])/(35*Sqrt[35])

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Rubi in Sympy [A]  time = 9.62118, size = 49, normalized size = 0.89 \[ - \frac{41 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1225} - \frac{13 \sqrt{3 x^{2} + 2}}{35 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-41*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/1225 - 13*sqrt(3*x
**2 + 2)/(35*(2*x + 3))

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Mathematica [A]  time = 0.093619, size = 70, normalized size = 1.27 \[ \frac{-\frac{455 \sqrt{3 x^2+2}}{2 x+3}-41 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+41 \sqrt{35} \log (2 x+3)}{1225} \]

Antiderivative was successfully verified.

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

[Out]

((-455*Sqrt[2 + 3*x^2])/(3 + 2*x) + 41*Sqrt[35]*Log[3 + 2*x] - 41*Sqrt[35]*Log[2
*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/1225

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Maple [A]  time = 0.016, size = 53, normalized size = 1. \[ -{\frac{13}{70}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{41\,\sqrt{35}}{1225}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/70/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)-41/1225*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 = 0.762794, size = 72, normalized size = 1.31 \[ \frac{41}{1225} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{35 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.278601, size = 108, normalized size = 1.96 \[ \frac{\sqrt{35}{\left (41 \,{\left (2 \, x + 3\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right ) - 26 \, \sqrt{35} \sqrt{3 \, x^{2} + 2}\right )}}{2450 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{4 x^{2} \sqrt{3 x^{2} + 2} + 12 x \sqrt{3 x^{2} + 2} + 9 \sqrt{3 x^{2} + 2}}\, dx - \int \left (- \frac{5}{4 x^{2} \sqrt{3 x^{2} + 2} + 12 x \sqrt{3 x^{2} + 2} + 9 \sqrt{3 x^{2} + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(4*x**2*sqrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2) + 9*sqrt(3*x**2 + 2
)), x) - Integral(-5/(4*x**2*sqrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2) + 9*sqrt(3
*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 2}{\left (2 \, x + 3\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^2), x)