3.1425 \(\int \frac{5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + (312 + 2137*x)/(7350*Sqrt[2 + 3*x^2]) - (1
04*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi [A]  time = 0.134243, antiderivative size = 73, 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 \[ \frac{41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac{2137 x+312}{7350 \sqrt{3 x^2+2}}-\frac{104 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + (312 + 2137*x)/(7350*Sqrt[2 + 3*x^2]) - (1
04*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi in Sympy [A]  time = 14.7675, size = 65, normalized size = 0.89 \[ \frac{123 x + 78}{630 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} + \frac{38466 x + 5616}{132300 \sqrt{3 x^{2} + 2}} - \frac{104 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{42875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(123*x + 78)/(630*(3*x**2 + 2)**(3/2)) + (38466*x + 5616)/(132300*sqrt(3*x**2 +
2)) - 104*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/42875

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Mathematica [A]  time = 0.208853, size = 78, normalized size = 1.07 \[ \frac{-624 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+\frac{35 \left (6411 x^3+936 x^2+5709 x+1534\right )}{\left (3 x^2+2\right )^{3/2}}+624 \sqrt{35} \log (2 x+3)}{257250} \]

Antiderivative was successfully verified.

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

[Out]

((35*(1534 + 5709*x + 936*x^2 + 6411*x^3))/(2 + 3*x^2)^(3/2) + 624*Sqrt[35]*Log[
3 + 2*x] - 624*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/257250

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Maple [B]  time = 0.012, size = 122, normalized size = 1.7 \[ -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{12}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{13}{105} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{39\,x}{140} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1833\,x}{4900}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{52}{1225}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{104\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*x/(3*x^2+2)^(3/2)-1/12*x/(3*x^2+2)^(1/2)+13/105/(3*(x+3/2)^2-9*x-19/4)^(3/
2)+39/140*x/(3*(x+3/2)^2-9*x-19/4)^(3/2)+1833/4900*x/(3*(x+3/2)^2-9*x-19/4)^(1/2
)+52/1225/(3*(x+3/2)^2-9*x-19/4)^(1/2)-104/42875*35^(1/2)*arctanh(2/35*(4-9*x)*3
5^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 0.764446, size = 109, normalized size = 1.49 \[ \frac{104}{42875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{2137 \, x}{7350 \, \sqrt{3 \, x^{2} + 2}} + \frac{52}{1225 \, \sqrt{3 \, x^{2} + 2}} + \frac{41 \, x}{210 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{13}{105 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

104/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)
) + 2137/7350*x/sqrt(3*x^2 + 2) + 52/1225/sqrt(3*x^2 + 2) + 41/210*x/(3*x^2 + 2)
^(3/2) + 13/105/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.280413, size = 146, normalized size = 2. \[ \frac{\sqrt{35}{\left (\sqrt{35}{\left (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt{3 \, x^{2} + 2} + 312 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{257250 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/257250*sqrt(35)*(sqrt(35)*(6411*x^3 + 936*x^2 + 5709*x + 1534)*sqrt(3*x^2 + 2)
+ 312*(9*x^4 + 12*x^2 + 4)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2
+ 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(9*x^4 + 12*x^2 + 4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308635, size = 126, normalized size = 1.73 \[ \frac{104}{42875} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{3 \,{\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

104/42875*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))) + 1/7350*(3*((2137
*x + 312)*x + 1903)*x + 1534)/(3*x^2 + 2)^(3/2)