3.1416 \(\int \frac{5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{41 x+26}{70 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{1051 \sqrt{3 x^2+2}}{42875 (2 x+3)}-\frac{27 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{23 \sqrt{3 x^2+2}}{525 (2 x+3)^3}-\frac{3312 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (23*Sqrt[2 + 3*x^2])/(525*(3 + 2*
x)^3) - (27*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)^2) - (1051*Sqrt[2 + 3*x^2])/(42875*
(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42875*Sqrt[35
])

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Rubi [A]  time = 0.243201, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{41 x+26}{70 (2 x+3)^3 \sqrt{3 x^2+2}}-\frac{1051 \sqrt{3 x^2+2}}{42875 (2 x+3)}-\frac{27 \sqrt{3 x^2+2}}{1225 (2 x+3)^2}+\frac{23 \sqrt{3 x^2+2}}{525 (2 x+3)^3}-\frac{3312 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^3*Sqrt[2 + 3*x^2]) + (23*Sqrt[2 + 3*x^2])/(525*(3 + 2*
x)^3) - (27*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)^2) - (1051*Sqrt[2 + 3*x^2])/(42875*
(3 + 2*x)) - (3312*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42875*Sqrt[35
])

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Rubi in Sympy [A]  time = 23.2318, size = 112, normalized size = 0.89 \[ - \frac{3312 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1500625} - \frac{1051 \sqrt{3 x^{2} + 2}}{42875 \left (2 x + 3\right )} - \frac{27 \sqrt{3 x^{2} + 2}}{1225 \left (2 x + 3\right )^{2}} + \frac{123 x + 78}{210 \left (2 x + 3\right )^{3} \sqrt{3 x^{2} + 2}} + \frac{23 \sqrt{3 x^{2} + 2}}{525 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3312*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/1500625 - 1051*s
qrt(3*x**2 + 2)/(42875*(2*x + 3)) - 27*sqrt(3*x**2 + 2)/(1225*(2*x + 3)**2) + (1
23*x + 78)/(210*(2*x + 3)**3*sqrt(3*x**2 + 2)) + 23*sqrt(3*x**2 + 2)/(525*(2*x +
3)**3)

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Mathematica [A]  time = 0.184346, size = 90, normalized size = 0.71 \[ \frac{-19872 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \left (75672 x^4+261036 x^3+237930 x^2+23349 x+29438\right )}{(2 x+3)^3 \sqrt{3 x^2+2}}+19872 \sqrt{35} \log (2 x+3)}{9003750} \]

Antiderivative was successfully verified.

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

[Out]

((-35*(29438 + 23349*x + 237930*x^2 + 261036*x^3 + 75672*x^4))/((3 + 2*x)^3*Sqrt
[2 + 3*x^2]) + 19872*Sqrt[35]*Log[3 + 2*x] - 19872*Sqrt[35]*Log[2*(4 - 9*x + Sqr
t[35]*Sqrt[2 + 3*x^2])])/9003750

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Maple [A]  time = 0.016, size = 128, normalized size = 1. \[ -{\frac{13}{840} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{17}{700} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{101}{2450} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{1656}{42875}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{3153\,x}{85750}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{3312\,\sqrt{35}}{1500625}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/840/(x+3/2)^3/(3*(x+3/2)^2-9*x-19/4)^(1/2)-17/700/(x+3/2)^2/(3*(x+3/2)^2-9*x
-19/4)^(1/2)-101/2450/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)+1656/42875/(3*(x+3/2)
^2-9*x-19/4)^(1/2)-3153/85750*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)-3312/1500625*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 = 0.759593, size = 248, normalized size = 1.97 \[ \frac{3312}{1500625} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{3153 \, x}{85750 \, \sqrt{3 \, x^{2} + 2}} + \frac{1656}{42875 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{105 \,{\left (8 \, \sqrt{3 \, x^{2} + 2} x^{3} + 36 \, \sqrt{3 \, x^{2} + 2} x^{2} + 54 \, \sqrt{3 \, x^{2} + 2} x + 27 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{17}{175 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{101}{1225 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3312/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x +
3)) - 3153/85750*x/sqrt(3*x^2 + 2) + 1656/42875/sqrt(3*x^2 + 2) - 13/105/(8*sqr
t(3*x^2 + 2)*x^3 + 36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 + 2)*x + 27*sqrt(3*x^2
+ 2)) - 17/175/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2
)) - 101/1225/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 0.281032, size = 188, normalized size = 1.49 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (75672 \, x^{4} + 261036 \, x^{3} + 237930 \, x^{2} + 23349 \, x + 29438\right )} \sqrt{3 \, x^{2} + 2} - 9936 \,{\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{9003750 \,{\left (24 \, x^{5} + 108 \, x^{4} + 178 \, x^{3} + 153 \, x^{2} + 108 \, x + 54\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9003750*sqrt(35)*(sqrt(35)*(75672*x^4 + 261036*x^3 + 237930*x^2 + 23349*x + 2
9438)*sqrt(3*x^2 + 2) - 9936*(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)
*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12
*x + 9)))/(24*x^5 + 108*x^4 + 178*x^3 + 153*x^2 + 108*x + 54)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.325339, size = 329, normalized size = 2.61 \[ \frac{3312}{1500625} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{3 \,{\left (10281 \, x - 12674\right )}}{3001250 \, \sqrt{3 \, x^{2} + 2}} - \frac{2 \,{\left (38949 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 253320 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 894510 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1481160 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 1275420 \, \sqrt{3} x - 106016 \, \sqrt{3} - 1275420 \, \sqrt{3 \, x^{2} + 2}\right )}}{1500625 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3312/1500625*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))) - 3/3001250*(10
281*x - 12674)/sqrt(3*x^2 + 2) - 2/1500625*(38949*(sqrt(3)*x - sqrt(3*x^2 + 2))^
5 + 253320*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 894510*(sqrt(3)*x - sqrt(3*
x^2 + 2))^3 - 1481160*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 1275420*sqrt(3)*
x - 106016*sqrt(3) - 1275420*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