3.1378 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=109 \[ -\frac{13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{171500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{85750 \sqrt{35}} \]

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2
)^(3/2))/(4900*(3 + 2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) - (1107*A
rcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(85750*Sqrt[35])

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Rubi [A]  time = 0.131769, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}-\frac{41 (4-9 x) \left (3 x^2+2\right )^{3/2}}{4900 (2 x+3)^4}-\frac{369 (4-9 x) \sqrt{3 x^2+2}}{171500 (2 x+3)^2}-\frac{1107 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{85750 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-369*(4 - 9*x)*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)^2) - (41*(4 - 9*x)*(2 + 3*x^2
)^(3/2))/(4900*(3 + 2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) - (1107*A
rcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(85750*Sqrt[35])

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Rubi in Sympy [A]  time = 18.456, size = 104, normalized size = 0.95 \[ - \frac{369 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{343000 \left (2 x + 3\right )^{2}} - \frac{41 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{9800 \left (2 x + 3\right )^{4}} - \frac{1107 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{3001250} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{175 \left (2 x + 3\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-369*(-18*x + 8)*sqrt(3*x**2 + 2)/(343000*(2*x + 3)**2) - 41*(-18*x + 8)*(3*x**2
+ 2)**(3/2)/(9800*(2*x + 3)**4) - 1107*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*s
qrt(3*x**2 + 2)))/3001250 - 13*(3*x**2 + 2)**(5/2)/(175*(2*x + 3)**5)

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Mathematica [A]  time = 0.118812, size = 90, normalized size = 0.83 \[ \frac{-2214 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (10602 x^4-189543 x^3+26682 x^2-64493 x+125252\right )}{(2 x+3)^5}+2214 \sqrt{35} \log (2 x+3)}{6002500} \]

Antiderivative was successfully verified.

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(125252 - 64493*x + 26682*x^2 - 189543*x^3 + 10602*x^4))/(
3 + 2*x)^5 + 2214*Sqrt[35]*Log[3 + 2*x] - 2214*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35
]*Sqrt[2 + 3*x^2])])/6002500

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Maple [B]  time = 0.019, size = 203, normalized size = 1.9 \[ -{\frac{13}{5600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{41}{39200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{369}{686000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{3813}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{43173}{210087500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{1476}{52521875} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{9963\,x}{6002500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{1107}{3001250}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{1107\,\sqrt{35}}{3001250}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{129519\,x}{210087500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/5600/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(5/2)-41/39200/(x+3/2)^4*(3*(x+3/2)^2-
9*x-19/4)^(5/2)-369/686000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(5/2)-3813/12005000/
(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-43173/210087500/(x+3/2)*(3*(x+3/2)^2-9*x-
19/4)^(5/2)+1476/52521875*(3*(x+3/2)^2-9*x-19/4)^(3/2)+9963/6002500*x*(3*(x+3/2)
^2-9*x-19/4)^(1/2)+1107/3001250*(12*(x+3/2)^2-36*x-19)^(1/2)-1107/3001250*35^(1/
2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+129519/210087500*
x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 0.786767, size = 282, normalized size = 2.59 \[ \frac{11439}{12005000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{175 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{41 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{2450 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{369 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{85750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{3813 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{3001250 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{9963}{6002500} \, \sqrt{3 \, x^{2} + 2} x + \frac{1107}{3001250} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{1107}{1500625} \, \sqrt{3 \, x^{2} + 2} - \frac{43173 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{12005000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11439/12005000*(3*x^2 + 2)^(3/2) - 13/175*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 +
720*x^3 + 1080*x^2 + 810*x + 243) - 41/2450*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 +
216*x^2 + 216*x + 81) - 369/85750*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27
) - 3813/3001250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 9963/6002500*sqrt(3*x^2
+ 2)*x + 1107/3001250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/
abs(2*x + 3)) + 1107/1500625*sqrt(3*x^2 + 2) - 43173/12005000*(3*x^2 + 2)^(3/2)/
(2*x + 3)

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Fricas [A]  time = 0.295486, size = 188, normalized size = 1.72 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (10602 \, x^{4} - 189543 \, x^{3} + 26682 \, x^{2} - 64493 \, x + 125252\right )} \sqrt{3 \, x^{2} + 2} - 1107 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{6002500 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6002500*sqrt(35)*(sqrt(35)*(10602*x^4 - 189543*x^3 + 26682*x^2 - 64493*x + 12
5252)*sqrt(3*x^2 + 2) - 1107*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 24
3)*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*x**2+2)**(3/2)/(3+2*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304848, size = 429, normalized size = 3.94 \[ \frac{1107}{3001250} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9 \,{\left (89686 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 138886 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 1224478 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 245133 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1224531 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 4374874 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 4855928 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 1339152 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 586816 \, \sqrt{3} x - 37696 \, \sqrt{3} + 586816 \, \sqrt{3 \, x^{2} + 2}\right )}}{2744000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1107/3001250*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))) - 9/2744000*(89
686*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 138886*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2
))^8 + 1224478*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 245133*sqrt(3)*(sqrt(3)*x - sqr
t(3*x^2 + 2))^6 - 1224531*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4374874*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 2))^4 + 4855928*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1339152*
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 586816*sqrt(3)*x - 37696*sqrt(3) + 586
816*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2)) - 2)^5