3.2553 \(\int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\)

Optimal. Leaf size=38 \[ 12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

[Out]

12*ArcTanh[Sqrt[3 + 2*x]] - (34*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15]

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Rubi [A]  time = 0.0827002, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ 12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

12*ArcTanh[Sqrt[3 + 2*x]] - (34*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15]

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Rubi in Sympy [A]  time = 18.3299, size = 36, normalized size = 0.95 \[ - \frac{34 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3*x**2+5*x+2)/(3+2*x)**(1/2),x)

[Out]

-34*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/15 + 12*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.0462219, size = 56, normalized size = 1.47 \[ -6 \log \left (1-\sqrt{2 x+3}\right )+6 \log \left (\sqrt{2 x+3}+1\right )-\frac{34 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

(-34*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15] - 6*Log[1 - Sqrt[3 + 2*x]] + 6*L
og[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.014, size = 44, normalized size = 1.2 \[ -6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{34\,\sqrt{15}}{15}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+5*x+2)/(3+2*x)^(1/2),x)

[Out]

-6*ln(-1+(3+2*x)^(1/2))-34/15*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+6*ln(
1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.791593, size = 82, normalized size = 2.16 \[ \frac{17}{15} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/15*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) +
6*log(sqrt(2*x + 3) + 1) - 6*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.289083, size = 89, normalized size = 2.34 \[ \frac{1}{15} \, \sqrt{15}{\left (6 \, \sqrt{15} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \sqrt{15} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 17 \, \log \left (\frac{\sqrt{15}{\left (3 \, x + 7\right )} - 15 \, \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*sqrt(15)*(6*sqrt(15)*log(sqrt(2*x + 3) + 1) - 6*sqrt(15)*log(sqrt(2*x + 3)
- 1) + 17*log((sqrt(15)*(3*x + 7) - 15*sqrt(2*x + 3))/(3*x + 2)))

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Sympy [A]  time = 8.2361, size = 95, normalized size = 2.5 \[ 34 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} > \frac{3}{5} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} < \frac{3}{5} \end{cases}\right ) - 6 \log{\left (-1 + \frac{1}{\sqrt{2 x + 3}} \right )} + 6 \log{\left (1 + \frac{1}{\sqrt{2 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)/(3*x**2+5*x+2)/(3+2*x)**(1/2),x)

[Out]

34*Piecewise((-sqrt(15)*acoth(sqrt(15)/(3*sqrt(2*x + 3)))/15, 1/(2*x + 3) > 3/5)
, (-sqrt(15)*atanh(sqrt(15)/(3*sqrt(2*x + 3)))/15, 1/(2*x + 3) < 3/5)) - 6*log(-
1 + 1/sqrt(2*x + 3)) + 6*log(1 + 1/sqrt(2*x + 3))

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GIAC/XCAS [A]  time = 0.278722, size = 88, normalized size = 2.32 \[ \frac{17}{15} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/15*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x
+ 3))) + 6*ln(sqrt(2*x + 3) + 1) - 6*ln(abs(sqrt(2*x + 3) - 1))