3.2569 \(\int \frac{(5-x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=102 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{3 \sqrt{2 x+3} (1063 x+878)}{10 \left (3 x^2+5 x+2\right )}+730 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4713}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-(Sqrt[3 + 2*x]*(29 + 35*x))/(2*(2 + 5*x + 3*x^2)^2) + (3*Sqrt[3 + 2*x]*(878 + 1
063*x))/(10*(2 + 5*x + 3*x^2)) + 730*ArcTanh[Sqrt[3 + 2*x]] - (4713*Sqrt[3/5]*Ar
cTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/5

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Rubi [A]  time = 0.188945, antiderivative size = 102, 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 \[ -\frac{\sqrt{2 x+3} (35 x+29)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{3 \sqrt{2 x+3} (1063 x+878)}{10 \left (3 x^2+5 x+2\right )}+730 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{4713}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(29 + 35*x))/(2*(2 + 5*x + 3*x^2)^2) + (3*Sqrt[3 + 2*x]*(878 + 1
063*x))/(10*(2 + 5*x + 3*x^2)) + 730*ArcTanh[Sqrt[3 + 2*x]] - (4713*Sqrt[3/5]*Ar
cTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/5

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Rubi in Sympy [A]  time = 34.0815, size = 87, normalized size = 0.85 \[ - \frac{\sqrt{2 x + 3} \left (35 x + 29\right )}{2 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{\sqrt{2 x + 3} \left (3189 x + 2634\right )}{10 \left (3 x^{2} + 5 x + 2\right )} - \frac{4713 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{25} + 730 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**(1/2)/(3*x**2+5*x+2)**3,x)

[Out]

-sqrt(2*x + 3)*(35*x + 29)/(2*(3*x**2 + 5*x + 2)**2) + sqrt(2*x + 3)*(3189*x + 2
634)/(10*(3*x**2 + 5*x + 2)) - 4713*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/25
+ 730*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.259762, size = 100, normalized size = 0.98 \[ \frac{\sqrt{2 x+3} \left (9567 x^3+23847 x^2+19373 x+5123\right )}{10 \left (3 x^2+5 x+2\right )^2}-365 \log \left (1-\sqrt{2 x+3}\right )+365 \log \left (\sqrt{2 x+3}+1\right )-\frac{4713}{5} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 2*x]*(5123 + 19373*x + 23847*x^2 + 9567*x^3))/(10*(2 + 5*x + 3*x^2)^2)
- (4713*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/5 - 365*Log[1 - Sqrt[3 + 2*
x]] + 365*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.029, size = 124, normalized size = 1.2 \[ 3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+56\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-365\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +162\,{\frac{1}{ \left ( 4+6\,x \right ) ^{2}} \left ({\frac{503\, \left ( 3+2\,x \right ) ^{3/2}}{90}}-{\frac{179\,\sqrt{3+2\,x}}{18}} \right ) }-{\frac{4713\,\sqrt{15}}{25}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+56\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+365\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^(1/2)/(3*x^2+5*x+2)^3,x)

[Out]

3/(-1+(3+2*x)^(1/2))^2+56/(-1+(3+2*x)^(1/2))-365*ln(-1+(3+2*x)^(1/2))+162*(503/9
0*(3+2*x)^(3/2)-179/18*(3+2*x)^(1/2))/(4+6*x)^2-4713/25*arctanh(1/5*15^(1/2)*(3+
2*x)^(1/2))*15^(1/2)-3/(1+(3+2*x)^(1/2))^2+56/(1+(3+2*x)^(1/2))+365*ln(1+(3+2*x)
^(1/2))

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Maxima [A]  time = 0.790626, size = 181, normalized size = 1.77 \[ \frac{4713}{50} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{9567 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 38409 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 49637 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 20555 \, \sqrt{2 \, x + 3}}{5 \,{\left (9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 365 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 365 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4713/50*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
+ 1/5*(9567*(2*x + 3)^(7/2) - 38409*(2*x + 3)^(5/2) + 49637*(2*x + 3)^(3/2) - 2
0555*sqrt(2*x + 3))/(9*(2*x + 3)^4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160*x - 2
15) + 365*log(sqrt(2*x + 3) + 1) - 365*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.288641, size = 243, normalized size = 2.38 \[ \frac{\sqrt{5}{\left (3650 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 3650 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 4713 \, \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (9567 \, x^{3} + 23847 \, x^{2} + 19373 \, x + 5123\right )} \sqrt{2 \, x + 3}\right )}}{50 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/50*sqrt(5)*(3650*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3
) + 1) - 3650*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3) - 1
) + 4713*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log((sqrt(5)*(3*x + 7) - 5
*sqrt(3)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(5)*(9567*x^3 + 23847*x^2 + 19373*x + 5
123)*sqrt(2*x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 100.604, size = 388, normalized size = 3.8 \[ - 2712 \left (\begin{cases} \frac{\sqrt{15} \left (- \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1\right )}\right )}{75} & \text{for}\: x \geq - \frac{3}{2} \wedge x < - \frac{2}{3} \end{cases}\right ) + 2040 \left (\begin{cases} \frac{\sqrt{15} \left (\frac{3 \log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text{for}\: x \geq - \frac{3}{2} \wedge x < - \frac{2}{3} \end{cases}\right ) + 2526 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right ) - 365 \log{\left (\sqrt{2 x + 3} - 1 \right )} + 365 \log{\left (\sqrt{2 x + 3} + 1 \right )} + \frac{56}{\sqrt{2 x + 3} + 1} - \frac{3}{\left (\sqrt{2 x + 3} + 1\right )^{2}} + \frac{56}{\sqrt{2 x + 3} - 1} + \frac{3}{\left (\sqrt{2 x + 3} - 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**(1/2)/(3*x**2+5*x+2)**3,x)

[Out]

-2712*Piecewise((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*s
qrt(2*x + 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) - 1/(4*(sqrt(15)*sq
rt(2*x + 3)/5 - 1)))/75, (x >= -3/2) & (x < -2/3))) + 2040*Piecewise((sqrt(15)*(
3*log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/16 - 3*log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/16
+ 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) + 1/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)**
2) + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1)) - 1/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1
)**2))/375, (x >= -3/2) & (x < -2/3))) + 2526*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)) - 365*log(sqrt(2*x + 3) - 1) + 365*log(sqrt(2*x + 3) + 1)
+ 56/(sqrt(2*x + 3) + 1) - 3/(sqrt(2*x + 3) + 1)**2 + 56/(sqrt(2*x + 3) - 1) + 3
/(sqrt(2*x + 3) - 1)**2

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GIAC/XCAS [A]  time = 0.27194, size = 162, normalized size = 1.59 \[ \frac{4713}{50} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{9567 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 38409 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 49637 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 20555 \, \sqrt{2 \, x + 3}}{5 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 365 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 365 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4713/50*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*
x + 3))) + 1/5*(9567*(2*x + 3)^(7/2) - 38409*(2*x + 3)^(5/2) + 49637*(2*x + 3)^(
3/2) - 20555*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)^2 + 365*ln(sqrt(2*x + 3)
+ 1) - 365*ln(abs(sqrt(2*x + 3) - 1))