3.2560 \(\int \frac{(5-x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x]] + (
316*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15]

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Rubi [A]  time = 0.128118, antiderivative size = 66, 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 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) - 82*ArcTanh[Sqrt[3 + 2*x]] + (
316*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/Sqrt[15]

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Rubi in Sympy [A]  time = 26.5932, size = 60, normalized size = 0.91 \[ - \frac{\sqrt{2 x + 3} \left (35 x + 29\right )}{3 x^{2} + 5 x + 2} + \frac{316 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} - 82 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2*x + 3)*(35*x + 29)/(3*x**2 + 5*x + 2) + 316*sqrt(15)*atanh(sqrt(15)*sqrt
(2*x + 3)/5)/15 - 82*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.120833, size = 84, normalized size = 1.27 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{3 x^2+5 x+2}+41 \log \left (1-\sqrt{2 x+3}\right )-41 \log \left (\sqrt{2 x+3}+1\right )+\frac{316 \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{\sqrt{15}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3 + 2*x]*(29 + 35*x))/(2 + 5*x + 3*x^2)) + (316*ArcTanh[Sqrt[3/5]*Sqrt[3
+ 2*x]])/Sqrt[15] + 41*Log[1 - Sqrt[3 + 2*x]] - 41*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.027, size = 86, normalized size = 1.3 \[ -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+41\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{34}{3}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{316\,\sqrt{15}}{15}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-41\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/(-1+(3+2*x)^(1/2))+41*ln(-1+(3+2*x)^(1/2))-34/3*(3+2*x)^(1/2)/(4/3+2*x)+316/1
5*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-6/(1+(3+2*x)^(1/2))-41*ln(1+(3+2*
x)^(1/2))

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Maxima [A]  time = 0.792018, size = 132, normalized size = 2. \[ -\frac{158}{15} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (35 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 47 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 41 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-158/15*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
- 2*(35*(2*x + 3)^(3/2) - 47*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 41*lo
g(sqrt(2*x + 3) + 1) + 41*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.288074, size = 167, normalized size = 2.53 \[ -\frac{\sqrt{15}{\left (41 \, \sqrt{15}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 41 \, \sqrt{15}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + \sqrt{15}{\left (35 \, x + 29\right )} \sqrt{2 \, x + 3} - 158 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{15}{\left (3 \, x + 7\right )} + 15 \, \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )}}{15 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(15)*(41*sqrt(15)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) - 41*sqrt(1
5)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) + sqrt(15)*(35*x + 29)*sqrt(2*x + 3)
- 158*(3*x^2 + 5*x + 2)*log((sqrt(15)*(3*x + 7) + 15*sqrt(2*x + 3))/(3*x + 2)))
/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 41.4695, size = 212, normalized size = 3.21 \[ 340 \left (\begin{cases} \frac{\sqrt{15} \left (- \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1\right )}\right )}{75} & \text{for}\: x \geq - \frac{3}{2} \wedge x < - \frac{2}{3} \end{cases}\right ) - 282 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right ) + 41 \log{\left (\sqrt{2 x + 3} - 1 \right )} - 41 \log{\left (\sqrt{2 x + 3} + 1 \right )} - \frac{6}{\sqrt{2 x + 3} + 1} - \frac{6}{\sqrt{2 x + 3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

340*Piecewise((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqr
t(2*x + 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) - 1/(4*(sqrt(15)*sqrt
(2*x + 3)/5 - 1)))/75, (x >= -3/2) & (x < -2/3))) - 282*Piecewise((-sqrt(15)*aco
th(sqrt(15)*sqrt(2*x + 3)/5)/15, 2*x + 3 > 5/3), (-sqrt(15)*atanh(sqrt(15)*sqrt(
2*x + 3)/5)/15, 2*x + 3 < 5/3)) + 41*log(sqrt(2*x + 3) - 1) - 41*log(sqrt(2*x +
3) + 1) - 6/(sqrt(2*x + 3) + 1) - 6/(sqrt(2*x + 3) - 1)

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GIAC/XCAS [A]  time = 0.269511, size = 138, normalized size = 2.09 \[ -\frac{158}{15} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{2 \,{\left (35 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 47 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 41 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-158/15*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*
x + 3))) - 2*(35*(2*x + 3)^(3/2) - 47*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)
- 41*ln(sqrt(2*x + 3) + 1) + 41*ln(abs(sqrt(2*x + 3) - 1))