3.1407 \(\int \frac{5-x}{(3+2 x)^6 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{10023 \sqrt{3 x^2+2}}{15006250 (2 x+3)}-\frac{1611 \sqrt{3 x^2+2}}{428750 (2 x+3)^2}-\frac{797 \sqrt{3 x^2+2}}{61250 (2 x+3)^3}-\frac{439 \sqrt{3 x^2+2}}{12250 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+2}}{175 (2 x+3)^5}+\frac{19737 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(175*(3 + 2*x)^5) - (439*Sqrt[2 + 3*x^2])/(12250*(3 + 2*x)
^4) - (797*Sqrt[2 + 3*x^2])/(61250*(3 + 2*x)^3) - (1611*Sqrt[2 + 3*x^2])/(428750
*(3 + 2*x)^2) - (10023*Sqrt[2 + 3*x^2])/(15006250*(3 + 2*x)) + (19737*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

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Rubi [A]  time = 0.298615, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{10023 \sqrt{3 x^2+2}}{15006250 (2 x+3)}-\frac{1611 \sqrt{3 x^2+2}}{428750 (2 x+3)^2}-\frac{797 \sqrt{3 x^2+2}}{61250 (2 x+3)^3}-\frac{439 \sqrt{3 x^2+2}}{12250 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+2}}{175 (2 x+3)^5}+\frac{19737 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{7503125 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 3*x^2]),x]

[Out]

(-13*Sqrt[2 + 3*x^2])/(175*(3 + 2*x)^5) - (439*Sqrt[2 + 3*x^2])/(12250*(3 + 2*x)
^4) - (797*Sqrt[2 + 3*x^2])/(61250*(3 + 2*x)^3) - (1611*Sqrt[2 + 3*x^2])/(428750
*(3 + 2*x)^2) - (10023*Sqrt[2 + 3*x^2])/(15006250*(3 + 2*x)) + (19737*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7503125*Sqrt[35])

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Rubi in Sympy [A]  time = 27.2391, size = 129, normalized size = 0.9 \[ \frac{19737 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{262609375} - \frac{10023 \sqrt{3 x^{2} + 2}}{15006250 \left (2 x + 3\right )} - \frac{1611 \sqrt{3 x^{2} + 2}}{428750 \left (2 x + 3\right )^{2}} - \frac{797 \sqrt{3 x^{2} + 2}}{61250 \left (2 x + 3\right )^{3}} - \frac{439 \sqrt{3 x^{2} + 2}}{12250 \left (2 x + 3\right )^{4}} - \frac{13 \sqrt{3 x^{2} + 2}}{175 \left (2 x + 3\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**6/(3*x**2+2)**(1/2),x)

[Out]

19737*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/262609375 - 1002
3*sqrt(3*x**2 + 2)/(15006250*(2*x + 3)) - 1611*sqrt(3*x**2 + 2)/(428750*(2*x + 3
)**2) - 797*sqrt(3*x**2 + 2)/(61250*(2*x + 3)**3) - 439*sqrt(3*x**2 + 2)/(12250*
(2*x + 3)**4) - 13*sqrt(3*x**2 + 2)/(175*(2*x + 3)**5)

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Mathematica [A]  time = 0.199712, size = 90, normalized size = 0.63 \[ \frac{19737 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (80184 x^4+706644 x^3+2487944 x^2+4314244 x+3409859\right )}{(2 x+3)^5}-19737 \sqrt{35} \log (2 x+3)}{262609375} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 3*x^2]),x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(3409859 + 4314244*x + 2487944*x^2 + 706644*x^3 + 80184*x^
4))/(3 + 2*x)^5 - 19737*Sqrt[35]*Log[3 + 2*x] + 19737*Sqrt[35]*Log[2*(4 - 9*x +
Sqrt[35]*Sqrt[2 + 3*x^2])])/262609375

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Maple [A]  time = 0.017, size = 137, normalized size = 1. \[ -{\frac{13}{5600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{439}{196000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{797}{490000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1611}{1715000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{10023}{30012500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{19737\,\sqrt{35}}{262609375}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(2*x+3)^6/(3*x^2+2)^(1/2),x)

[Out]

-13/5600/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(1/2)-439/196000/(x+3/2)^4*(3*(x+3/2)^
2-9*x-19/4)^(1/2)-797/490000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(1/2)-1611/1715000
/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-10023/30012500/(x+3/2)*(3*(x+3/2)^2-9*x-
19/4)^(1/2)+19737/262609375*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 = 0.771335, size = 236, normalized size = 1.65 \[ -\frac{19737}{262609375} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{175 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{439 \, \sqrt{3 \, x^{2} + 2}}{12250 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{797 \, \sqrt{3 \, x^{2} + 2}}{61250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1611 \, \sqrt{3 \, x^{2} + 2}}{428750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{10023 \, \sqrt{3 \, x^{2} + 2}}{15006250 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^6),x, algorithm="maxima")

[Out]

-19737/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2
*x + 3)) - 13/175*sqrt(3*x^2 + 2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x
+ 243) - 439/12250*sqrt(3*x^2 + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 7
97/61250*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1611/428750*sqrt(3*x^2 +
2)/(4*x^2 + 12*x + 9) - 10023/15006250*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 0.281786, size = 189, normalized size = 1.32 \[ -\frac{\sqrt{35}{\left (2 \, \sqrt{35}{\left (80184 \, x^{4} + 706644 \, x^{3} + 2487944 \, x^{2} + 4314244 \, x + 3409859\right )} \sqrt{3 \, x^{2} + 2} - 19737 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} - 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{525218750 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^6),x, algorithm="fricas")

[Out]

-1/525218750*sqrt(35)*(2*sqrt(35)*(80184*x^4 + 706644*x^3 + 2487944*x^2 + 431424
4*x + 3409859)*sqrt(3*x^2 + 2) - 19737*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 +
810*x + 243)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) - 35*sqrt(3*x^2 + 2)*(9*x - 4))
/(4*x^2 + 12*x + 9)))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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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)**6/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.341006, size = 429, normalized size = 3. \[ -\frac{19737}{262609375} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{3 \,{\left (26316 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 355266 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 5320218 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 11098773 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} + 6945939 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 49794206 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 76607832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 16740688 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 5232096 \, \sqrt{3} x + 213824 \, \sqrt{3} + 5232096 \, \sqrt{3 \, x^{2} + 2}\right )}}{30012500 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^6),x, algorithm="giac")

[Out]

-19737/262609375*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
0*(26316*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 355266*sqrt(3)*(sqrt(3)*x - sqrt(3*x^
2 + 2))^8 + 5320218*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 11098773*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2))^6 + 6945939*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 49794206*sqrt
(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 76607832*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 +
16740688*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 5232096*sqrt(3)*x + 213824*s
qrt(3) + 5232096*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)^5