3.2551 \(\int \frac{(5-x) (3+2 x)^{3/2}}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2}{9} (2 x+3)^{3/2}+\frac{62}{9} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{170}{9} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(62*Sqrt[3 + 2*x])/9 - (2*(3 + 2*x)^(3/2))/9 + 12*ArcTanh[Sqrt[3 + 2*x]] - (170*
Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/9

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Rubi [A]  time = 0.175495, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{2}{9} (2 x+3)^{3/2}+\frac{62}{9} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{170}{9} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(62*Sqrt[3 + 2*x])/9 - (2*(3 + 2*x)^(3/2))/9 + 12*ArcTanh[Sqrt[3 + 2*x]] - (170*
Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/9

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Rubi in Sympy [A]  time = 33.5093, size = 60, normalized size = 0.88 \[ - \frac{2 \left (2 x + 3\right )^{\frac{3}{2}}}{9} + \frac{62 \sqrt{2 x + 3}}{9} - \frac{170 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{27} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(3/2)/9 + 62*sqrt(2*x + 3)/9 - 170*sqrt(15)*atanh(sqrt(15)*sqrt(2*
x + 3)/5)/27 + 12*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.0646801, size = 82, normalized size = 1.21 \[ -\frac{2}{27} \left (3 (2 x+3)^{3/2}-93 \sqrt{2 x+3}+81 \log \left (1-\sqrt{2 x+3}\right )-81 \log \left (\sqrt{2 x+3}+1\right )+85 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-93*Sqrt[3 + 2*x] + 3*(3 + 2*x)^(3/2) + 85*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[
3 + 2*x]] + 81*Log[1 - Sqrt[3 + 2*x]] - 81*Log[1 + Sqrt[3 + 2*x]]))/27

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Maple [A]  time = 0.016, size = 62, normalized size = 0.9 \[ -{\frac{2}{9} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{62}{9}\sqrt{3+2\,x}}-6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{170\,\sqrt{15}}{27}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(3+2*x)^(3/2)+62/9*(3+2*x)^(1/2)-6*ln(-1+(3+2*x)^(1/2))-170/27*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 = 0.792904, size = 107, normalized size = 1.57 \[ -\frac{2}{9} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{85}{27} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{62}{9} \, \sqrt{2 \, x + 3} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(2*x + 3)^(3/2) + 85/27*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15
) + 3*sqrt(2*x + 3))) + 62/9*sqrt(2*x + 3) + 6*log(sqrt(2*x + 3) + 1) - 6*log(sq
rt(2*x + 3) - 1)

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Fricas [A]  time = 0.289395, size = 117, normalized size = 1.72 \[ -\frac{1}{27} \, \sqrt{3}{\left (4 \, \sqrt{3} \sqrt{2 \, x + 3}{\left (x - 14\right )} - 54 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 54 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 85 \, \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} - 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )} \]

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),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(4*sqrt(3)*sqrt(2*x + 3)*(x - 14) - 54*sqrt(3)*log(sqrt(2*x + 3) +
1) + 54*sqrt(3)*log(sqrt(2*x + 3) - 1) - 85*sqrt(5)*log((sqrt(3)*(3*x + 7) - 3*
sqrt(5)*sqrt(2*x + 3))/(3*x + 2)))

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Sympy [A]  time = 33.8711, size = 114, normalized size = 1.68 \[ - \frac{2 \left (2 x + 3\right )^{\frac{3}{2}}}{9} + \frac{62 \sqrt{2 x + 3}}{9} + \frac{850 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right )}{9} - 6 \log{\left (\sqrt{2 x + 3} - 1 \right )} + 6 \log{\left (\sqrt{2 x + 3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(3/2)/9 + 62*sqrt(2*x + 3)/9 + 850*Piecewise((-sqrt(15)*acoth(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))/9 - 6*log(sqrt(2*x + 3) - 1) + 6*log(sqrt(2*x + 3) + 1)

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GIAC/XCAS [A]  time = 0.272761, size = 112, normalized size = 1.65 \[ -\frac{2}{9} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{85}{27} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{62}{9} \, \sqrt{2 \, x + 3} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(2*x + 3)^(3/2) + 85/27*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/
(sqrt(15) + 3*sqrt(2*x + 3))) + 62/9*sqrt(2*x + 3) + 6*ln(sqrt(2*x + 3) + 1) - 6
*ln(abs(sqrt(2*x + 3) - 1))