3.2561 \(\int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}-58 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{384}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(-3*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) - 58*ArcTanh[Sqrt[3 + 2*x]]
+ (384*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/5

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Rubi [A]  time = 0.133051, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}-58 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{384}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) - 58*ArcTanh[Sqrt[3 + 2*x]]
+ (384*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/5

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Rubi in Sympy [A]  time = 26.5473, size = 60, normalized size = 0.83 \[ - \frac{\sqrt{2 x + 3} \left (141 x + 111\right )}{5 \left (3 x^{2} + 5 x + 2\right )} + \frac{384 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{25} - 58 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2*x + 3)*(141*x + 111)/(5*(3*x**2 + 5*x + 2)) + 384*sqrt(15)*atanh(sqrt(15
)*sqrt(2*x + 3)/5)/25 - 58*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.126133, size = 90, normalized size = 1.25 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}+29 \log \left (1-\sqrt{2 x+3}\right )-29 \log \left (\sqrt{2 x+3}+1\right )+\frac{384}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[3 + 2*x]*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) + (384*Sqrt[3/5]*ArcTanh[Sq
rt[3/5]*Sqrt[3 + 2*x]])/5 + 29*Log[1 - Sqrt[3 + 2*x]] - 29*Log[1 + Sqrt[3 + 2*x]
]

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Maple [A]  time = 0.026, size = 86, normalized size = 1.2 \[ -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+29\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{34}{5}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{384\,\sqrt{15}}{25}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-29\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/(-1+(3+2*x)^(1/2))+29*ln(-1+(3+2*x)^(1/2))-34/5*(3+2*x)^(1/2)/(4/3+2*x)+384/2
5*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-6/(1+(3+2*x)^(1/2))-29*ln(1+(3+2*
x)^(1/2))

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Maxima [A]  time = 0.793716, size = 132, normalized size = 1.83 \[ -\frac{192}{25} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{6 \,{\left (47 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 67 \, \sqrt{2 \, x + 3}\right )}}{5 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 29 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 29 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-192/25*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
- 6/5*(47*(2*x + 3)^(3/2) - 67*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 29*
log(sqrt(2*x + 3) + 1) + 29*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.29102, size = 177, normalized size = 2.46 \[ -\frac{\sqrt{5}{\left (145 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 145 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 192 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} + 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + 3 \, \sqrt{5}{\left (47 \, x + 37\right )} \sqrt{2 \, x + 3}\right )}}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*sqrt(5)*(145*sqrt(5)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) - 145*sqrt(5
)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 192*sqrt(3)*(3*x^2 + 5*x + 2)*log((
sqrt(5)*(3*x + 7) + 5*sqrt(3)*sqrt(2*x + 3))/(3*x + 2)) + 3*sqrt(5)*(47*x + 37)*
sqrt(2*x + 3))/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 45.5471, size = 221, normalized size = 3.07 \[ - \frac{612 \left (\begin{cases} \frac{\sqrt{15} \left (- \frac{\log{\left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{4} + \frac{\log{\left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{4} - \frac{1}{4 \left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )} - \frac{1}{4 \left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )}\right )}{45} & \text{for}\: x > - \frac{2}{3} \end{cases}\right )}{5} - \frac{1254 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} > \frac{3}{5} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} < \frac{3}{5} \end{cases}\right )}{5} + 29 \log{\left (-1 + \frac{1}{\sqrt{2 x + 3}} \right )} - 29 \log{\left (1 + \frac{1}{\sqrt{2 x + 3}} \right )} + \frac{6}{1 + \frac{1}{\sqrt{2 x + 3}}} + \frac{6}{-1 + \frac{1}{\sqrt{2 x + 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-612*Piecewise((sqrt(15)*(-log(-1 + sqrt(15)/(3*sqrt(2*x + 3)))/4 + log(1 + sqrt
(15)/(3*sqrt(2*x + 3)))/4 - 1/(4*(1 + sqrt(15)/(3*sqrt(2*x + 3)))) - 1/(4*(-1 +
sqrt(15)/(3*sqrt(2*x + 3)))))/45, x > -2/3))/5 - 1254*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))/5 + 29*log(-1 + 1/sqrt(2*x + 3)) - 29*
log(1 + 1/sqrt(2*x + 3)) + 6/(1 + 1/sqrt(2*x + 3)) + 6/(-1 + 1/sqrt(2*x + 3))

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GIAC/XCAS [A]  time = 0.272075, size = 138, normalized size = 1.92 \[ -\frac{192}{25} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{6 \,{\left (47 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 67 \, \sqrt{2 \, x + 3}\right )}}{5 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 29 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 29 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-192/25*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*
x + 3))) - 6/5*(47*(2*x + 3)^(3/2) - 67*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 1
9) - 29*ln(sqrt(2*x + 3) + 1) + 29*ln(abs(sqrt(2*x + 3) - 1))