3.2503 \(\int \frac{5-x}{(3+2 x)^5 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{681 \sqrt{3 x^2+5 x+2}}{250 (2 x+3)}-\frac{41 \sqrt{3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac{86 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac{5771 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2000 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) - (86*Sqrt[2 + 5*x + 3*x^2])/(75*(3
+ 2*x)^3) - (41*Sqrt[2 + 5*x + 3*x^2])/(24*(3 + 2*x)^2) - (681*Sqrt[2 + 5*x + 3
*x^2])/(250*(3 + 2*x)) + (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(2000*Sqrt[5])

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Rubi [A]  time = 0.300287, antiderivative size = 139, 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 \[ -\frac{681 \sqrt{3 x^2+5 x+2}}{250 (2 x+3)}-\frac{41 \sqrt{3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac{86 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac{5771 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2000 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) - (86*Sqrt[2 + 5*x + 3*x^2])/(75*(3
+ 2*x)^3) - (41*Sqrt[2 + 5*x + 3*x^2])/(24*(3 + 2*x)^2) - (681*Sqrt[2 + 5*x + 3
*x^2])/(250*(3 + 2*x)) + (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2
])])/(2000*Sqrt[5])

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Rubi in Sympy [A]  time = 38.0384, size = 129, normalized size = 0.93 \[ - \frac{5771 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{10000} - \frac{681 \sqrt{3 x^{2} + 5 x + 2}}{250 \left (2 x + 3\right )} - \frac{41 \sqrt{3 x^{2} + 5 x + 2}}{24 \left (2 x + 3\right )^{2}} - \frac{86 \sqrt{3 x^{2} + 5 x + 2}}{75 \left (2 x + 3\right )^{3}} - \frac{13 \sqrt{3 x^{2} + 5 x + 2}}{20 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5771*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/10000 - 681*
sqrt(3*x**2 + 5*x + 2)/(250*(2*x + 3)) - 41*sqrt(3*x**2 + 5*x + 2)/(24*(2*x + 3)
**2) - 86*sqrt(3*x**2 + 5*x + 2)/(75*(2*x + 3)**3) - 13*sqrt(3*x**2 + 5*x + 2)/(
20*(2*x + 3)**4)

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Mathematica [A]  time = 0.144392, size = 90, normalized size = 0.65 \[ \frac{-17313 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (65376 x^3+314692 x^2+509668 x+279039\right )}{(2 x+3)^4}+17313 \sqrt{5} \log (2 x+3)}{30000} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(279039 + 509668*x + 314692*x^2 + 65376*x^3))/(3 + 2
*x)^4 + 17313*Sqrt[5]*Log[3 + 2*x] - 17313*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt
[2 + 5*x + 3*x^2]])/30000

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Maple [A]  time = 0.016, size = 116, normalized size = 0.8 \[ -{\frac{13}{320}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{43}{300}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{41}{96}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{681}{500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{5771\,\sqrt{5}}{10000}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(1/2)-43/300/(x+3/2)^3*(3*(x+3/2)^2-4*x
-19/4)^(1/2)-41/96/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(1/2)-681/500/(x+3/2)*(3*(x+
3/2)^2-4*x-19/4)^(1/2)-5771/10000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+
3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 0.806759, size = 212, normalized size = 1.53 \[ -\frac{5771}{10000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{86 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{75 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{41 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{24 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{681 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{250 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5771/10000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x
+ 3) - 2) - 13/20*sqrt(3*x^2 + 5*x + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81
) - 86/75*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 41/24*sqrt(3*x^2
+ 5*x + 2)/(4*x^2 + 12*x + 9) - 681/250*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 0.287388, size = 176, normalized size = 1.27 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (65376 \, x^{3} + 314692 \, x^{2} + 509668 \, x + 279039\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 17313 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{60000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60000*sqrt(5)*(4*sqrt(5)*(65376*x^3 + 314692*x^2 + 509668*x + 279039)*sqrt(3*
x^2 + 5*x + 2) - 17313*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((sqrt(5)*(12
4*x^2 + 212*x + 89) + 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(
16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{32 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 810 x \sqrt{3 x^{2} + 5 x + 2} + 243 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{32 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 810 x \sqrt{3 x^{2} + 5 x + 2} + 243 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) +
720*x**3*sqrt(3*x**2 + 5*x + 2) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(
3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(32*x**5*sqrt(
3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 720*x**3*sqrt(3*x**2 + 5*x
+ 2) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(3*x**2 + 5*x + 2) + 243*sq
rt(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^5), x)