3.2559 \(\int \frac{(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}-106 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{248}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-(Sqrt[3 + 2*x]*(121 + 139*x))/(3*(2 + 5*x + 3*x^2)) - 106*ArcTanh[Sqrt[3 + 2*x]
] + (248*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3

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Rubi [A]  time = 0.132949, 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 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}-106 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{248}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(121 + 139*x))/(3*(2 + 5*x + 3*x^2)) - 106*ArcTanh[Sqrt[3 + 2*x]
] + (248*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/3

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Rubi in Sympy [A]  time = 26.8443, size = 60, normalized size = 0.83 \[ - \frac{\sqrt{2 x + 3} \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + \frac{248 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{9} - 106 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2*x + 3)*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 248*sqrt(15)*atanh(sqrt(15
)*sqrt(2*x + 3)/5)/9 - 106*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.119445, size = 88, normalized size = 1.22 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{9 x^2+15 x+6}+53 \log \left (1-\sqrt{2 x+3}\right )-53 \log \left (\sqrt{2 x+3}+1\right )+\frac{248}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[3 + 2*x]*(121 + 139*x))/(6 + 15*x + 9*x^2)) + (248*Sqrt[5/3]*ArcTanh[Sqr
t[3/5]*Sqrt[3 + 2*x]])/3 + 53*Log[1 - Sqrt[3 + 2*x]] - 53*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.025, size = 86, normalized size = 1.2 \[ -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+53\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{170}{9}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{248\,\sqrt{15}}{9}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-53\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/(-1+(3+2*x)^(1/2))+53*ln(-1+(3+2*x)^(1/2))-170/9*(3+2*x)^(1/2)/(4/3+2*x)+248/
9*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-6/(1+(3+2*x)^(1/2))-53*ln(1+(3+2*
x)^(1/2))

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Maxima [A]  time = 0.791946, size = 132, normalized size = 1.83 \[ -\frac{124}{9} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (139 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 175 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 53 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-124/9*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
- 2/3*(139*(2*x + 3)^(3/2) - 175*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 53
*log(sqrt(2*x + 3) + 1) + 53*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.290357, size = 176, normalized size = 2.44 \[ -\frac{\sqrt{3}{\left (159 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 159 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 124 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} + 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{3}{\left (139 \, x + 121\right )} \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(159*sqrt(3)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) - 159*sqrt(3)
*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 124*sqrt(5)*(3*x^2 + 5*x + 2)*log((s
qrt(3)*(3*x + 7) + 3*sqrt(5)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(3)*(139*x + 121)*s
qrt(2*x + 3))/(3*x^2 + 5*x + 2)

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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)**(3/2)/(3*x**2+5*x+2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270386, size = 138, normalized size = 1.92 \[ -\frac{124}{9} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{2 \,{\left (139 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 175 \, \sqrt{2 \, x + 3}\right )}}{3 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 53 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-124/9*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x
+ 3))) - 2/3*(139*(2*x + 3)^(3/2) - 175*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x -
19) - 53*ln(sqrt(2*x + 3) + 1) + 53*ln(abs(sqrt(2*x + 3) - 1))