3.1379 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=131 \[ -\frac{29 \left (3 x^2+2\right )^{5/2}}{1750 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}-\frac{(4-9 x) \left (3 x^2+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{9 (4-9 x) \sqrt{3 x^2+2}}{17500 (2 x+3)^2}-\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8750 \sqrt{35}} \]

[Out]

(-9*(4 - 9*x)*Sqrt[2 + 3*x^2])/(17500*(3 + 2*x)^2) - ((4 - 9*x)*(2 + 3*x^2)^(3/2
))/(500*(3 + 2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(210*(3 + 2*x)^6) - (29*(2 + 3*x^2
)^(5/2))/(1750*(3 + 2*x)^5) - (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])
/(8750*Sqrt[35])

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Rubi [A]  time = 0.188687, antiderivative size = 131, 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 \[ -\frac{29 \left (3 x^2+2\right )^{5/2}}{1750 (2 x+3)^5}-\frac{13 \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}-\frac{(4-9 x) \left (3 x^2+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{9 (4-9 x) \sqrt{3 x^2+2}}{17500 (2 x+3)^2}-\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8750 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^7,x]

[Out]

(-9*(4 - 9*x)*Sqrt[2 + 3*x^2])/(17500*(3 + 2*x)^2) - ((4 - 9*x)*(2 + 3*x^2)^(3/2
))/(500*(3 + 2*x)^4) - (13*(2 + 3*x^2)^(5/2))/(210*(3 + 2*x)^6) - (29*(2 + 3*x^2
)^(5/2))/(1750*(3 + 2*x)^5) - (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])
/(8750*Sqrt[35])

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Rubi in Sympy [A]  time = 22.8475, size = 122, normalized size = 0.93 \[ - \frac{9 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{35000 \left (2 x + 3\right )^{2}} - \frac{\left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{1000 \left (2 x + 3\right )^{4}} - \frac{27 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{306250} - \frac{29 \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{1750 \left (2 x + 3\right )^{5}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{210 \left (2 x + 3\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**7,x)

[Out]

-9*(-18*x + 8)*sqrt(3*x**2 + 2)/(35000*(2*x + 3)**2) - (-18*x + 8)*(3*x**2 + 2)*
*(3/2)/(1000*(2*x + 3)**4) - 27*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x*
*2 + 2)))/306250 - 29*(3*x**2 + 2)**(5/2)/(1750*(2*x + 3)**5) - 13*(3*x**2 + 2)*
*(5/2)/(210*(2*x + 3)**6)

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Mathematica [A]  time = 0.155184, size = 95, normalized size = 0.73 \[ \frac{-162 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (432 x^5+2160 x^4-39195 x^3+33180 x^2+3675 x+39748\right )}{(2 x+3)^6}+162 \sqrt{35} \log (2 x+3)}{1837500} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^7,x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(39748 + 3675*x + 33180*x^2 - 39195*x^3 + 2160*x^4 + 432*x
^5))/(3 + 2*x)^6 + 162*Sqrt[35]*Log[3 + 2*x] - 162*Sqrt[35]*Log[2*(4 - 9*x + Sqr
t[35]*Sqrt[2 + 3*x^2])])/1837500

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Maple [B]  time = 0.022, size = 224, normalized size = 1.7 \[ -{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{29}{56000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{1}{4000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{9}{70000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{93}{1225000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{1053}{21437500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{36}{5359375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{243\,x}{612500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{27}{306250}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{27\,\sqrt{35}}{306250}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{3159\,x}{21437500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^7,x)

[Out]

-13/13440/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(5/2)-29/56000/(x+3/2)^5*(3*(x+3/2)^2
-9*x-19/4)^(5/2)-1/4000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(5/2)-9/70000/(x+3/2)^3
*(3*(x+3/2)^2-9*x-19/4)^(5/2)-93/1225000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(5/2)-
1053/21437500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)+36/5359375*(3*(x+3/2)^2-9*x-1
9/4)^(3/2)+243/612500*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+27/306250*(12*(x+3/2)^2-36*
x-19)^(1/2)-27/306250*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-
19)^(1/2))+3159/21437500*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 0.785184, size = 340, normalized size = 2.6 \[ \frac{279}{1225000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{29 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{1750 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{250 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{8750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{93 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{306250 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{243}{612500} \, \sqrt{3 \, x^{2} + 2} x + \frac{27}{306250} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{27}{153125} \, \sqrt{3 \, x^{2} + 2} - \frac{1053 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{1225000 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^7,x, algorithm="maxima")

[Out]

279/1225000*(3*x^2 + 2)^(3/2) - 13/210*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 216
0*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 29/1750*(3*x^2 + 2)^(5/2)/(32*x^5
+ 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1/250*(3*x^2 + 2)^(5/2)/(16*x^4
+ 96*x^3 + 216*x^2 + 216*x + 81) - 9/8750*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54
*x + 27) - 93/306250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 243/612500*sqrt(3*x^
2 + 2)*x + 27/306250*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/a
bs(2*x + 3)) + 27/153125*sqrt(3*x^2 + 2) - 1053/1225000*(3*x^2 + 2)^(3/2)/(2*x +
3)

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Fricas [A]  time = 0.297238, size = 208, normalized size = 1.59 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (432 \, x^{5} + 2160 \, x^{4} - 39195 \, x^{3} + 33180 \, x^{2} + 3675 \, x + 39748\right )} \sqrt{3 \, x^{2} + 2} - 81 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{1837500 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^7,x, algorithm="fricas")

[Out]

-1/1837500*sqrt(35)*(sqrt(35)*(432*x^5 + 2160*x^4 - 39195*x^3 + 33180*x^2 + 3675
*x + 39748)*sqrt(3*x^2 + 2) - 81*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*
x^2 + 2916*x + 729)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*
x - 4))/(4*x^2 + 12*x + 9)))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2
+ 2916*x + 729)

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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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.316238, size = 490, normalized size = 3.74 \[ \frac{27}{306250} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9 \,{\left (96 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 5959 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 4120 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 8620 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 225240 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 57988 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 648336 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 213680 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 309440 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 45040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 10752 \, \sqrt{3} x + 512 \, \sqrt{3} + 10752 \, \sqrt{3 \, x^{2} + 2}\right )}}{280000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(3/2)*(x - 5)/(2*x + 3)^7,x, algorithm="giac")

[Out]

27/306250*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))) - 9/280000*(96*(sq
rt(3)*x - sqrt(3*x^2 + 2))^11 + 5959*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 -
4120*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 8620*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)
)^8 - 225240*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 57988*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 2))^6 - 648336*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 213680*sqrt(3)*(sqrt(3)*
x - sqrt(3*x^2 + 2))^4 - 309440*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 45040*sqrt(3)*
(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 10752*sqrt(3)*x + 512*sqrt(3) + 10752*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)^6