3.2555 \(\int \frac{5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\)

Optimal. Leaf size=68 \[ -\frac{198}{25 \sqrt{2 x+3}}-\frac{26}{15 (2 x+3)^{3/2}}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{102}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-26/(15*(3 + 2*x)^(3/2)) - 198/(25*Sqrt[3 + 2*x]) + 12*ArcTanh[Sqrt[3 + 2*x]] -
(102*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi [A]  time = 0.182966, 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 \[ -\frac{198}{25 \sqrt{2 x+3}}-\frac{26}{15 (2 x+3)^{3/2}}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{102}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-26/(15*(3 + 2*x)^(3/2)) - 198/(25*Sqrt[3 + 2*x]) + 12*ArcTanh[Sqrt[3 + 2*x]] -
(102*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi in Sympy [A]  time = 34.0615, size = 60, normalized size = 0.88 \[ - \frac{102 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{125} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{198}{25 \sqrt{2 x + 3}} - \frac{26}{15 \left (2 x + 3\right )^{\frac{3}{2}}} \]

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),x)

[Out]

-102*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/125 + 12*atanh(sqrt(2*x + 3)) - 19
8/(25*sqrt(2*x + 3)) - 26/(15*(2*x + 3)**(3/2))

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Mathematica [A]  time = 0.0955261, size = 86, normalized size = 1.26 \[ -\frac{198}{25 \sqrt{2 x+3}}-\frac{26}{15 (2 x+3)^{3/2}}-6 \log \left (1-\sqrt{2 x+3}\right )+6 \log \left (\sqrt{2 x+3}+1\right )-\frac{102}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-26/(15*(3 + 2*x)^(3/2)) - 198/(25*Sqrt[3 + 2*x]) - (102*Sqrt[3/5]*ArcTanh[Sqrt[
3/5]*Sqrt[3 + 2*x]])/25 - 6*Log[1 - Sqrt[3 + 2*x]] + 6*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.02, size = 62, normalized size = 0.9 \[ -6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{102\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{26}{15} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{198}{25}{\frac{1}{\sqrt{3+2\,x}}}}+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*ln(-1+(3+2*x)^(1/2))-102/125*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-26/
15/(3+2*x)^(3/2)-198/25/(3+2*x)^(1/2)+6*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.788257, size = 101, normalized size = 1.49 \[ \frac{51}{125} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{4 \,{\left (297 \, x + 478\right )}}{75 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

51/125*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
- 4/75*(297*x + 478)/(2*x + 3)^(3/2) + 6*log(sqrt(2*x + 3) + 1) - 6*log(sqrt(2*x
+ 3) - 1)

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Fricas [A]  time = 0.291024, size = 149, normalized size = 2.19 \[ \frac{\sqrt{5}{\left (450 \, \sqrt{5}{\left (2 \, x + 3\right )}^{\frac{3}{2}} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 450 \, \sqrt{5}{\left (2 \, x + 3\right )}^{\frac{3}{2}} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 153 \, \sqrt{3}{\left (2 \, x + 3\right )}^{\frac{3}{2}} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) - 4 \, \sqrt{5}{\left (297 \, x + 478\right )}\right )}}{375 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/375*sqrt(5)*(450*sqrt(5)*(2*x + 3)^(3/2)*log(sqrt(2*x + 3) + 1) - 450*sqrt(5)*
(2*x + 3)^(3/2)*log(sqrt(2*x + 3) - 1) + 153*sqrt(3)*(2*x + 3)^(3/2)*log((sqrt(5
)*(3*x + 7) - 5*sqrt(3)*sqrt(2*x + 3))/(3*x + 2)) - 4*sqrt(5)*(297*x + 478))/(2*
x + 3)^(3/2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{12 x^{4} \sqrt{2 x + 3} + 56 x^{3} \sqrt{2 x + 3} + 95 x^{2} \sqrt{2 x + 3} + 69 x \sqrt{2 x + 3} + 18 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{5}{12 x^{4} \sqrt{2 x + 3} + 56 x^{3} \sqrt{2 x + 3} + 95 x^{2} \sqrt{2 x + 3} + 69 x \sqrt{2 x + 3} + 18 \sqrt{2 x + 3}}\right )\, dx \]

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),x)

[Out]

-Integral(x/(12*x**4*sqrt(2*x + 3) + 56*x**3*sqrt(2*x + 3) + 95*x**2*sqrt(2*x +
3) + 69*x*sqrt(2*x + 3) + 18*sqrt(2*x + 3)), x) - Integral(-5/(12*x**4*sqrt(2*x
+ 3) + 56*x**3*sqrt(2*x + 3) + 95*x**2*sqrt(2*x + 3) + 69*x*sqrt(2*x + 3) + 18*s
qrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.271524, size = 107, normalized size = 1.57 \[ \frac{51}{125} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{4 \,{\left (297 \, x + 478\right )}}{75 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

51/125*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x
+ 3))) - 4/75*(297*x + 478)/(2*x + 3)^(3/2) + 6*ln(sqrt(2*x + 3) + 1) - 6*ln(ab
s(sqrt(2*x + 3) - 1))