3.1414 \(\int \frac{5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{41 x+26}{70 (2 x+3) \sqrt{3 x^2+2}}+\frac{19 \sqrt{3 x^2+2}}{1225 (2 x+3)}-\frac{632 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (19*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x
)) - (632*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi [A]  time = 0.129757, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{41 x+26}{70 (2 x+3) \sqrt{3 x^2+2}}+\frac{19 \sqrt{3 x^2+2}}{1225 (2 x+3)}-\frac{632 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (19*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x
)) - (632*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi in Sympy [A]  time = 14.0449, size = 70, normalized size = 0.85 \[ - \frac{632 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{42875} + \frac{123 x + 78}{210 \left (2 x + 3\right ) \sqrt{3 x^{2} + 2}} + \frac{19 \sqrt{3 x^{2} + 2}}{1225 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-632*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/42875 + (123*x +
78)/(210*(2*x + 3)*sqrt(3*x**2 + 2)) + 19*sqrt(3*x**2 + 2)/(1225*(2*x + 3))

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Mathematica [A]  time = 0.136596, size = 80, normalized size = 0.98 \[ \frac{\frac{35 \left (114 x^2+1435 x+986\right )}{(2 x+3) \sqrt{3 x^2+2}}-1264 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+1264 \sqrt{35} \log (2 x+3)}{85750} \]

Antiderivative was successfully verified.

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

[Out]

((35*(986 + 1435*x + 114*x^2))/((3 + 2*x)*Sqrt[2 + 3*x^2]) + 1264*Sqrt[35]*Log[3
+ 2*x] - 1264*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/85750

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Maple [A]  time = 0.015, size = 86, normalized size = 1.1 \[ -{\frac{13}{70} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{316}{1225}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{57\,x}{2450}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{632\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/70/(x+3/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)+316/1225/(3*(x+3/2)^2-9*x-19/4)^(1/2
)+57/2450*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)-632/42875*35^(1/2)*arctanh(2/35*(4-9*x)
*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 0.759868, size = 116, normalized size = 1.41 \[ \frac{632}{42875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{57 \, x}{2450 \, \sqrt{3 \, x^{2} + 2}} + \frac{316}{1225 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{35 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

632/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)
) + 57/2450*x/sqrt(3*x^2 + 2) + 316/1225/sqrt(3*x^2 + 2) - 13/35/(2*sqrt(3*x^2 +
2)*x + 3*sqrt(3*x^2 + 2))

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Fricas [A]  time = 0.275446, size = 147, normalized size = 1.79 \[ \frac{\sqrt{35}{\left (\sqrt{35}{\left (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt{3 \, x^{2} + 2} + 632 \,{\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{85750 \,{\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/85750*sqrt(35)*(sqrt(35)*(114*x^2 + 1435*x + 986)*sqrt(3*x^2 + 2) + 632*(6*x^3
+ 9*x^2 + 4*x + 6)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(9*
x - 4))/(4*x^2 + 12*x + 9)))/(6*x^3 + 9*x^2 + 4*x + 6)

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x - 5)/((3*x^2 + 2)^(3/2)*(2*x + 3)^2), x)