3.2562 \(\int \frac{5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{3 (47 x+37)}{5 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}-\frac{506}{25 \sqrt{2 x+3}}-34 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{1356}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-506/(25*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(5*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)) -
34*ArcTanh[Sqrt[3 + 2*x]] + (1356*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi [A]  time = 0.187018, antiderivative size = 85, 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 \[ -\frac{3 (47 x+37)}{5 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}-\frac{506}{25 \sqrt{2 x+3}}-34 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{1356}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-506/(25*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(5*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)) -
34*ArcTanh[Sqrt[3 + 2*x]] + (1356*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi in Sympy [A]  time = 33.8598, size = 73, normalized size = 0.86 \[ \frac{1356 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{125} - 34 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{141 x + 111}{5 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )} - \frac{506}{25 \sqrt{2 x + 3}} \]

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]

1356*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/125 - 34*atanh(sqrt(2*x + 3)) - (1
41*x + 111)/(5*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)) - 506/(25*sqrt(2*x + 3))

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Mathematica [A]  time = 0.143242, size = 103, normalized size = 1.21 \[ -\frac{3 \sqrt{2 x+3} (201 x+151)}{25 \left (3 x^2+5 x+2\right )}-\frac{104}{25 \sqrt{2 x+3}}+17 \log \left (1-\sqrt{2 x+3}\right )-17 \log \left (\sqrt{2 x+3}+1\right )+\frac{1356}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-104/(25*Sqrt[3 + 2*x]) - (3*Sqrt[3 + 2*x]*(151 + 201*x))/(25*(2 + 5*x + 3*x^2))
+ (1356*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25 + 17*Log[1 - Sqrt[3 + 2*
x]] - 17*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.028, size = 95, normalized size = 1.1 \[ -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+17\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{102}{25}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{1356\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{104}{25}{\frac{1}{\sqrt{3+2\,x}}}}-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-17\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

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))+17*ln(-1+(3+2*x)^(1/2))-102/25*(3+2*x)^(1/2)/(4/3+2*x)+135
6/125*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-104/25/(3+2*x)^(1/2)-6/(1+(3+
2*x)^(1/2))-17*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.787962, size = 144, normalized size = 1.69 \[ -\frac{678}{125} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (759 \,{\left (2 \, x + 3\right )}^{2} - 2638 \, x - 3697\right )}}{25 \,{\left (3 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - 8 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + 5 \, \sqrt{2 \, x + 3}\right )}} - 17 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 17 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-678/125*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))
) - 2/25*(759*(2*x + 3)^2 - 2638*x - 3697)/(3*(2*x + 3)^(5/2) - 8*(2*x + 3)^(3/2
) + 5*sqrt(2*x + 3)) - 17*log(sqrt(2*x + 3) + 1) + 17*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.287526, size = 211, normalized size = 2.48 \[ -\frac{\sqrt{5}{\left (425 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 425 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 678 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} + 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (1518 \, x^{2} + 3235 \, x + 1567\right )}\right )}}{125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/125*sqrt(5)*(425*sqrt(5)*(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*log(sqrt(2*x + 3) +
1) - 425*sqrt(5)*(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*log(sqrt(2*x + 3) - 1) - 678*sq
rt(3)*(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*log((sqrt(5)*(3*x + 7) + 5*sqrt(3)*sqrt(2*
x + 3))/(3*x + 2)) + sqrt(5)*(1518*x^2 + 3235*x + 1567))/((3*x^2 + 5*x + 2)*sqrt
(2*x + 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{18 x^{5} \sqrt{2 x + 3} + 87 x^{4} \sqrt{2 x + 3} + 164 x^{3} \sqrt{2 x + 3} + 151 x^{2} \sqrt{2 x + 3} + 68 x \sqrt{2 x + 3} + 12 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{5}{18 x^{5} \sqrt{2 x + 3} + 87 x^{4} \sqrt{2 x + 3} + 164 x^{3} \sqrt{2 x + 3} + 151 x^{2} \sqrt{2 x + 3} + 68 x \sqrt{2 x + 3} + 12 \sqrt{2 x + 3}}\right )\, dx \]

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]

-Integral(x/(18*x**5*sqrt(2*x + 3) + 87*x**4*sqrt(2*x + 3) + 164*x**3*sqrt(2*x +
3) + 151*x**2*sqrt(2*x + 3) + 68*x*sqrt(2*x + 3) + 12*sqrt(2*x + 3)), x) - Inte
gral(-5/(18*x**5*sqrt(2*x + 3) + 87*x**4*sqrt(2*x + 3) + 164*x**3*sqrt(2*x + 3)
+ 151*x**2*sqrt(2*x + 3) + 68*x*sqrt(2*x + 3) + 12*sqrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.271215, size = 150, normalized size = 1.76 \[ -\frac{678}{125} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{2 \,{\left (759 \,{\left (2 \, x + 3\right )}^{2} - 2638 \, x - 3697\right )}}{25 \,{\left (3 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - 8 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + 5 \, \sqrt{2 \, x + 3}\right )}} - 17 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 17 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-678/125*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2
*x + 3))) - 2/25*(759*(2*x + 3)^2 - 2638*x - 3697)/(3*(2*x + 3)^(5/2) - 8*(2*x +
3)^(3/2) + 5*sqrt(2*x + 3)) - 17*ln(sqrt(2*x + 3) + 1) + 17*ln(abs(sqrt(2*x + 3
) - 1))