3.2510 \(\int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ \frac{26 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{5 \sqrt{5}}-\frac{6 (47 x+37)}{5 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-6*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (26*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sq
rt[2 + 5*x + 3*x^2])])/(5*Sqrt[5])

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Rubi [A]  time = 0.107146, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{26 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{5 \sqrt{5}}-\frac{6 (47 x+37)}{5 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (26*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sq
rt[2 + 5*x + 3*x^2])])/(5*Sqrt[5])

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Rubi in Sympy [A]  time = 17.8143, size = 58, normalized size = 0.94 \[ - \frac{2 \left (141 x + 111\right )}{5 \sqrt{3 x^{2} + 5 x + 2}} - \frac{26 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(141*x + 111)/(5*sqrt(3*x**2 + 5*x + 2)) - 26*sqrt(5)*atanh(sqrt(5)*(-8*x - 7
)/(10*sqrt(3*x**2 + 5*x + 2)))/25

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Mathematica [A]  time = 0.118204, size = 86, normalized size = 1.39 \[ \frac{2}{25} \left (\frac{-13 \sqrt{5} \sqrt{3 x^2+5 x+2} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-705 x-555}{\sqrt{3 x^2+5 x+2}}+13 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(13*Sqrt[5]*Log[3 + 2*x] + (-555 - 705*x - 13*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]*L
og[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]])/Sqrt[2 + 5*x + 3*x^2]))/25

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Maple [A]  time = 0.01, size = 87, normalized size = 1.4 \[{(5+6\,x){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{13}{5}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{260+312\,x}{5}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{26\,\sqrt{5}}{25}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5+6*x)/(3*x^2+5*x+2)^(1/2)+13/5/(3*(x+3/2)^2-4*x-19/4)^(1/2)-52/5*(5+6*x)/(3*(x
+3/2)^2-4*x-19/4)^(1/2)-26/25*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)
^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.787993, size = 97, normalized size = 1.56 \[ -\frac{26}{25} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{282 \, x}{5 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{222}{5 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-26/25*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3)
- 2) - 282/5*x/sqrt(3*x^2 + 5*x + 2) - 222/5/sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.281262, size = 135, normalized size = 2.18 \[ -\frac{\sqrt{5}{\left (6 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (47 \, x + 37\right )} - 13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*sqrt(5)*(6*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(47*x + 37) - 13*(3*x^2 + 5*x + 2
)*log((sqrt(5)*(124*x^2 + 212*x + 89) + 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x
^2 + 12*x + 9)))/(3*x^2 + 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{6 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{3 x^{2} + 5 x + 2} + 6 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{6 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{3 x^{2} + 5 x + 2} + 6 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(6*x**3*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(3*x**2 + 5*x + 2) + 19
*x*sqrt(3*x**2 + 5*x + 2) + 6*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(6*x**3*
sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(3*x**2 + 5*x + 2) + 19*x*sqrt(3*x**2 + 5*x
+ 2) + 6*sqrt(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.309028, size = 126, normalized size = 2.03 \[ \frac{26}{25} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{6 \,{\left (47 \, x + 37\right )}}{5 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

26/25*sqrt(5)*ln(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x +
2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 6/5*
(47*x + 37)/sqrt(3*x^2 + 5*x + 2)