3.1415 \(\int \frac{5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (9*Sqrt[2 + 3*x^2])/(245*(3 + 2*x
)^2) - (331*Sqrt[2 + 3*x^2])/(8575*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

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Rubi [A]  time = 0.190306, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{41 x+26}{70 (2 x+3)^2 \sqrt{3 x^2+2}}-\frac{331 \sqrt{3 x^2+2}}{8575 (2 x+3)}+\frac{9 \sqrt{3 x^2+2}}{245 (2 x+3)^2}-\frac{1962 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (9*Sqrt[2 + 3*x^2])/(245*(3 + 2*x
)^2) - (331*Sqrt[2 + 3*x^2])/(8575*(3 + 2*x)) - (1962*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

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Rubi in Sympy [A]  time = 18.7328, size = 92, normalized size = 0.88 \[ - \frac{1962 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{300125} - \frac{331 \sqrt{3 x^{2} + 2}}{8575 \left (2 x + 3\right )} + \frac{123 x + 78}{210 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 2}} + \frac{9 \sqrt{3 x^{2} + 2}}{245 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1962*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/300125 - 331*sqr
t(3*x**2 + 2)/(8575*(2*x + 3)) + (123*x + 78)/(210*(2*x + 3)**2*sqrt(3*x**2 + 2)
) + 9*sqrt(3*x**2 + 2)/(245*(2*x + 3)**2)

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Mathematica [A]  time = 0.132631, size = 85, normalized size = 0.82 \[ \frac{-3924 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+\frac{35 \left (-3972 x^3-4068 x^2+7397 x+3658\right )}{(2 x+3)^2 \sqrt{3 x^2+2}}+3924 \sqrt{35} \log (2 x+3)}{600250} \]

Antiderivative was successfully verified.

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

[Out]

((35*(3658 + 7397*x - 4068*x^2 - 3972*x^3))/((3 + 2*x)^2*Sqrt[2 + 3*x^2]) + 3924
*Sqrt[35]*Log[3 + 2*x] - 3924*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2]
)])/600250

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Maple [A]  time = 0.017, size = 107, normalized size = 1. \[ -{\frac{13}{280} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{103}{980} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{981}{8575}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{993\,x}{17150}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{1962\,\sqrt{35}}{300125}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/280/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(1/2)-103/980/(x+3/2)/(3*(x+3/2)^2-9*x-
19/4)^(1/2)+981/8575/(3*(x+3/2)^2-9*x-19/4)^(1/2)-993/17150*x/(3*(x+3/2)^2-9*x-1
9/4)^(1/2)-1962/300125*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 = 0.762532, size = 173, normalized size = 1.66 \[ \frac{1962}{300125} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{993 \, x}{17150 \, \sqrt{3 \, x^{2} + 2}} + \frac{981}{8575 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{70 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{103}{490 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1962/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x +
3)) - 993/17150*x/sqrt(3*x^2 + 2) + 981/8575/sqrt(3*x^2 + 2) - 13/70/(4*sqrt(3*x
^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 103/490/(2*sqrt(3*x^2
+ 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 0.277187, size = 167, normalized size = 1.61 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt{3 \, x^{2} + 2} - 1962 \,{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{600250 \,{\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/600250*sqrt(35)*(sqrt(35)*(3972*x^3 + 4068*x^2 - 7397*x - 3658)*sqrt(3*x^2 +
2) - 1962*(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)*log(-(sqrt(35)*(93*x^2 - 36*x +
43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(12*x^4 + 36*x^3 + 35*
x^2 + 24*x + 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317239, size = 269, normalized size = 2.59 \[ \frac{1962}{300125} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{3 \,{\left (157 \, x - 1478\right )}}{85750 \, \sqrt{3 \, x^{2} + 2}} - \frac{768 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt{3} x + 856 \, \sqrt{3} + 6168 \, \sqrt{3 \, x^{2} + 2}}{6125 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1962/300125*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))) - 3/85750*(157*x
- 1478)/sqrt(3*x^2 + 2) - 1/6125*(768*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 2461*sq
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6168*sqrt(3)*x + 856*sqrt(3) + 6168*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