3.1390 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=133 \[ -\frac{(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac{3 (1917 x+6125) \sqrt{3 x^2+2}}{448 (2 x+3)}-\frac{188379 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{896 \sqrt{35}}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(3*(6125 + 1917*x)*Sqrt[2 + 3*x^2])/(448*(3 + 2*x)) - ((5003 + 5517*x)*(2 + 3*x^
2)^(3/2))/(672*(3 + 2*x)^3) - ((19 + 4*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) -
(2625*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (188379*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/(896*Sqrt[35])

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Rubi [A]  time = 0.243281, antiderivative size = 133, 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 \[ -\frac{(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}-\frac{(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{672 (2 x+3)^3}+\frac{3 (1917 x+6125) \sqrt{3 x^2+2}}{448 (2 x+3)}-\frac{188379 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{896 \sqrt{35}}-\frac{2625}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(6125 + 1917*x)*Sqrt[2 + 3*x^2])/(448*(3 + 2*x)) - ((5003 + 5517*x)*(2 + 3*x^
2)^(3/2))/(672*(3 + 2*x)^3) - ((19 + 4*x)*(2 + 3*x^2)^(5/2))/(16*(3 + 2*x)^4) -
(2625*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (188379*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/(896*Sqrt[35])

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Rubi in Sympy [A]  time = 24.3425, size = 117, normalized size = 0.88 \[ - \frac{2625 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{128} - \frac{188379 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{31360} + \frac{\left (368064 x + 1176000\right ) \sqrt{3 x^{2} + 2}}{28672 \left (2 x + 3\right )} - \frac{\left (88272 x + 80048\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{10752 \left (2 x + 3\right )^{3}} - \frac{\left (8 x + 38\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{32 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2625*sqrt(3)*asinh(sqrt(6)*x/2)/128 - 188379*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)
/(35*sqrt(3*x**2 + 2)))/31360 + (368064*x + 1176000)*sqrt(3*x**2 + 2)/(28672*(2*
x + 3)) - (88272*x + 80048)*(3*x**2 + 2)**(3/2)/(10752*(2*x + 3)**3) - (8*x + 38
)*(3*x**2 + 2)**(5/2)/(32*(2*x + 3)**4)

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Mathematica [A]  time = 0.255411, size = 112, normalized size = 0.84 \[ \frac{-565137 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{70 \sqrt{3 x^2+2} \left (3024 x^5-57456 x^4-898734 x^3-2762820 x^2-3335009 x-1421955\right )}{(2 x+3)^4}+565137 \sqrt{35} \log (2 x+3)-1929375 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{94080} \]

Antiderivative was successfully verified.

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

[Out]

((-70*Sqrt[2 + 3*x^2]*(-1421955 - 3335009*x - 2762820*x^2 - 898734*x^3 - 57456*x
^4 + 3024*x^5))/(3 + 2*x)^4 - 1929375*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 565137*Sqrt
[35]*Log[3 + 2*x] - 565137*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])
/94080

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Maple [B]  time = 0.019, size = 227, normalized size = 1.7 \[ -{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}+{\frac{23}{117600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1041}{343000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{29717}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{188379}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{58629\,x}{274400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{58491\,x}{15680}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{2625\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{62793}{137200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{188379}{31360}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{188379\,\sqrt{35}}{31360}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }-{\frac{89151\,x}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)+23/117600/(x+3/2)^3*(3*(x+3/2)^2
-9*x-19/4)^(7/2)-1041/343000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+29717/600250
0/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+188379/6002500*(3*(x+3/2)^2-9*x-19/4)^(5/
2)-58629/274400*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-58491/15680*x*(3*(x+3/2)^2-9*x-19
/4)^(1/2)-2625/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)+62793/137200*(3*(x+3/2)^2-9*x-
19/4)^(3/2)+188379/31360*(12*(x+3/2)^2-36*x-19)^(1/2)-188379/31360*35^(1/2)*arct
anh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-89151/6002500*x*(3*(x+3/
2)^2-9*x-19/4)^(5/2)

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Maxima [A]  time = 0.777712, size = 278, normalized size = 2.09 \[ \frac{3123}{343000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac{23 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{14700 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1041 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{85750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{58629}{274400} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{62793}{137200} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{29717 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{343000 \,{\left (2 \, x + 3\right )}} - \frac{58491}{15680} \, \sqrt{3 \, x^{2} + 2} x - \frac{2625}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{188379}{31360} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{188379}{15680} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3123/343000*(3*x^2 + 2)^(5/2) - 13/140*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*
x^2 + 216*x + 81) + 23/14700*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 10
41/85750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 58629/274400*(3*x^2 + 2)^(3/2)*x
+ 62793/137200*(3*x^2 + 2)^(3/2) + 29717/343000*(3*x^2 + 2)^(5/2)/(2*x + 3) - 5
8491/15680*sqrt(3*x^2 + 2)*x - 2625/128*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 188379/
31360*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) +
188379/15680*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.293116, size = 250, normalized size = 1.88 \[ \frac{\sqrt{35}{\left (55125 \, \sqrt{35} \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - 4 \, \sqrt{35}{\left (3024 \, x^{5} - 57456 \, x^{4} - 898734 \, x^{3} - 2762820 \, x^{2} - 3335009 \, x - 1421955\right )} \sqrt{3 \, x^{2} + 2} + 565137 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{188160 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/188160*sqrt(35)*(55125*sqrt(35)*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 8
1)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - 4*sqrt(35)*(3024*x^5 - 57456*x^4
- 898734*x^3 - 2762820*x^2 - 3335009*x - 1421955)*sqrt(3*x^2 + 2) + 565137*(16*
x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sq
rt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x
+ 81)

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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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.514092, size = 594, normalized size = 4.47 \[ -\frac{188379}{31360} \, \sqrt{35}{\rm ln}\left (\sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} - 9\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + \frac{2625}{128} \, \sqrt{3}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{10752} \,{\left (\frac{7 \,{\left (\frac{35 \,{\left (\frac{1365 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 2129 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 57681 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 242979 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac{9 \,{\left (256 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{3}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 93 \, \sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 582 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + 225 \, \sqrt{35}{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{64 \,{\left ({\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-188379/31360*sqrt(35)*ln(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + s
qrt(35)/(2*x + 3)) - 9)*sign(1/(2*x + 3)) + 2625/128*sqrt(3)*ln(1/2*abs(-2*sqrt(
3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(sqrt(3)
+ sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sign(1/(2*x +
3)) - 1/10752*(7*(35*(1365*sign(1/(2*x + 3))/(2*x + 3) - 2129*sign(1/(2*x + 3))
)/(2*x + 3) + 57681*sign(1/(2*x + 3)))/(2*x + 3) - 242979*sign(1/(2*x + 3)))*sqr
t(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 9/64*(256*(sqrt(-18/(2*x + 3) + 35/(2*x
+ 3)^2 + 3) + sqrt(35)/(2*x + 3))^3*sign(1/(2*x + 3)) - 93*sqrt(35)*(sqrt(-18/(2
*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2*sign(1/(2*x + 3)) - 582*(s
qrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sign(1/(2*x + 3))
+ 225*sqrt(35)*sign(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + s
qrt(35)/(2*x + 3))^2 - 3)^2