3.2558 \(\int \frac{(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt{2 x+3}-130 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

30*Sqrt[3 + 2*x] - ((3 + 2*x)^(3/2)*(121 + 139*x))/(3*(2 + 5*x + 3*x^2)) - 130*A
rcTanh[Sqrt[3 + 2*x]] + 100*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

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Rubi [A]  time = 0.182865, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt{2 x+3}-130 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

30*Sqrt[3 + 2*x] - ((3 + 2*x)^(3/2)*(121 + 139*x))/(3*(2 + 5*x + 3*x^2)) - 130*A
rcTanh[Sqrt[3 + 2*x]] + 100*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

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Rubi in Sympy [A]  time = 34.0264, size = 70, normalized size = 0.86 \[ - \frac{\left (2 x + 3\right )^{\frac{3}{2}} \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + 30 \sqrt{2 x + 3} + \frac{100 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{3} - 130 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(3/2)*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 30*sqrt(2*x + 3) + 100*
sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/3 - 130*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.133036, size = 101, normalized size = 1.25 \[ -\frac{\sqrt{2 x+3} (587 x+533)}{9 \left (3 x^2+5 x+2\right )}-\frac{8}{9} \sqrt{2 x+3}+65 \log \left (1-\sqrt{2 x+3}\right )-65 \log \left (\sqrt{2 x+3}+1\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-8*Sqrt[3 + 2*x])/9 - (Sqrt[3 + 2*x]*(533 + 587*x))/(9*(2 + 5*x + 3*x^2)) + 100
*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]] + 65*Log[1 - Sqrt[3 + 2*x]] - 65*Log
[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.028, size = 95, normalized size = 1.2 \[ -{\frac{8}{9}\sqrt{3+2\,x}}-6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+65\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{850}{27}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{100\,\sqrt{15}}{3}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-65\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9*(3+2*x)^(1/2)-6/(-1+(3+2*x)^(1/2))+65*ln(-1+(3+2*x)^(1/2))-850/27*(3+2*x)^(
1/2)/(4/3+2*x)+100/3*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-6/(1+(3+2*x)^(
1/2))-65*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.78745, size = 144, normalized size = 1.78 \[ -\frac{50}{3} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{8}{9} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (587 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 695 \, \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 65 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-50/3*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) -
8/9*sqrt(2*x + 3) - 2/9*(587*(2*x + 3)^(3/2) - 695*sqrt(2*x + 3))/(3*(2*x + 3)^
2 - 16*x - 19) - 65*log(sqrt(2*x + 3) + 1) + 65*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.288726, size = 182, normalized size = 2.25 \[ -\frac{\sqrt{3}{\left (195 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 195 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 150 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} + 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{3}{\left (8 \, x^{2} + 209 \, x + 183\right )} \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(195*sqrt(3)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) - 195*sqrt(3)
*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 150*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)*(8*x^2 + 209*x
+ 183)*sqrt(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)**(5/2)/(3*x**2+5*x+2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280268, size = 150, normalized size = 1.85 \[ -\frac{50}{3} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{8}{9} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (587 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 695 \, \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 65 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-50/3*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x
+ 3))) - 8/9*sqrt(2*x + 3) - 2/9*(587*(2*x + 3)^(3/2) - 695*sqrt(2*x + 3))/(3*(2
*x + 3)^2 - 16*x - 19) - 65*ln(sqrt(2*x + 3) + 1) + 65*ln(abs(sqrt(2*x + 3) - 1)
)