3.1388 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=126 \[ -\frac{(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (29 x+178) \left (3 x^2+2\right )^{3/2}}{32 (2 x+3)}+\frac{15}{64} (859-267 x) \sqrt{3 x^2+2}-\frac{12885}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(15*(859 - 267*x)*Sqrt[2 + 3*x^2])/64 + (5*(178 + 29*x)*(2 + 3*x^2)^(3/2))/(32*(
3 + 2*x)) - ((29 + 2*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^2) - (43995*Sqrt[3]*Arc
Sinh[Sqrt[3/2]*x])/128 - (12885*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*
x^2])])/128

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Rubi [A]  time = 0.235875, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(2 x+29) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (29 x+178) \left (3 x^2+2\right )^{3/2}}{32 (2 x+3)}+\frac{15}{64} (859-267 x) \sqrt{3 x^2+2}-\frac{12885}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{43995}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(15*(859 - 267*x)*Sqrt[2 + 3*x^2])/64 + (5*(178 + 29*x)*(2 + 3*x^2)^(3/2))/(32*(
3 + 2*x)) - ((29 + 2*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^2) - (43995*Sqrt[3]*Arc
Sinh[Sqrt[3/2]*x])/128 - (12885*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*
x^2])])/128

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Rubi in Sympy [A]  time = 23.6369, size = 114, normalized size = 0.9 \[ \frac{5 \left (- 153792 x + 494784\right ) \sqrt{3 x^{2} + 2}}{12288} - \frac{43995 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{128} - \frac{12885 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{128} + \frac{5 \left (696 x + 4272\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{768 \left (2 x + 3\right )} - \frac{\left (4 x + 58\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{32 \left (2 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-153792*x + 494784)*sqrt(3*x**2 + 2)/12288 - 43995*sqrt(3)*asinh(sqrt(6)*x/2)
/128 - 12885*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/128 + 5*(
696*x + 4272)*(3*x**2 + 2)**(3/2)/(768*(2*x + 3)) - (4*x + 58)*(3*x**2 + 2)**(5/
2)/(32*(2*x + 3)**2)

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Mathematica [A]  time = 0.2602, size = 112, normalized size = 0.89 \[ \frac{1}{128} \left (-12885 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{2 \sqrt{3 x^2+2} \left (72 x^5-696 x^4+2826 x^3-19268 x^2-127403 x-126181\right )}{(2 x+3)^2}+12885 \sqrt{35} \log (2 x+3)-43995 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[2 + 3*x^2]*(-126181 - 127403*x - 19268*x^2 + 2826*x^3 - 696*x^4 + 72*x
^5))/(3 + 2*x)^2 - 43995*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 12885*Sqrt[35]*Log[3 + 2
*x] - 12885*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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Maple [A]  time = 0.016, size = 185, normalized size = 1.5 \[ -{\frac{13}{280} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{421}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{2577}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{807\,x}{224} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{4005\,x}{64}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{43995\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{859}{112} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{12885}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{12885\,\sqrt{35}}{128}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }-{\frac{1263\,x}{4900} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/280/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+421/4900/(x+3/2)*(3*(x+3/2)^2-9*x
-19/4)^(7/2)+2577/4900*(3*(x+3/2)^2-9*x-19/4)^(5/2)-807/224*x*(3*(x+3/2)^2-9*x-1
9/4)^(3/2)-4005/64*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)-43995/128*arcsinh(1/2*x*6^(1/2
))*3^(1/2)+859/112*(3*(x+3/2)^2-9*x-19/4)^(3/2)+12885/128*(12*(x+3/2)^2-36*x-19)
^(1/2)-12885/128*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(
1/2))-1263/4900*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [A]  time = 0.784382, size = 196, normalized size = 1.56 \[ \frac{39}{280} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{807}{224} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{859}{112} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{421 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{280 \,{\left (2 \, x + 3\right )}} - \frac{4005}{64} \, \sqrt{3 \, x^{2} + 2} x - \frac{43995}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{12885}{128} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{12885}{64} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39/280*(3*x^2 + 2)^(5/2) - 13/70*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 807/224*
(3*x^2 + 2)^(3/2)*x + 859/112*(3*x^2 + 2)^(3/2) + 421/280*(3*x^2 + 2)^(5/2)/(2*x
+ 3) - 4005/64*sqrt(3*x^2 + 2)*x - 43995/128*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1
2885/128*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3))
+ 12885/64*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.292963, size = 197, normalized size = 1.56 \[ \frac{43995 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12885 \, \sqrt{35}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \,{\left (72 \, x^{5} - 696 \, x^{4} + 2826 \, x^{3} - 19268 \, x^{2} - 127403 \, x - 126181\right )} \sqrt{3 \, x^{2} + 2}}{256 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*(43995*sqrt(3)*(4*x^2 + 12*x + 9)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 -
1) + 12885*sqrt(35)*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4)
+ 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 4*(72*x^5 - 696*x^4 + 2826*x^3 - 192
68*x^2 - 127403*x - 126181)*sqrt(3*x^2 + 2))/(4*x^2 + 12*x + 9)

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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*x**2+2)**(5/2)/(3+2*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.332209, size = 311, normalized size = 2.47 \[ -\frac{1}{32} \,{\left (3 \,{\left ({\left (3 \, x - 38\right )} x + 225\right )} x - 4177\right )} \sqrt{3 \, x^{2} + 2} + \frac{43995}{128} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{12885}{128} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{35 \,{\left (11472 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 25829 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 57912 \, \sqrt{3} x + 8984 \, \sqrt{3} + 57912 \, \sqrt{3 \, x^{2} + 2}\right )}}{256 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(3*((3*x - 38)*x + 225)*x - 4177)*sqrt(3*x^2 + 2) + 43995/128*sqrt(3)*ln(-
sqrt(3)*x + sqrt(3*x^2 + 2)) + 12885/128*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))) + 35/256*(11472*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 25829*sqrt(3)*(sq
rt(3)*x - sqrt(3*x^2 + 2))^2 - 57912*sqrt(3)*x + 8984*sqrt(3) + 57912*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