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

Optimal. Leaf size=63 \[ \frac{121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}-\frac{22}{125} \log \left (5 x^2+3 x+2\right )+\frac{4 x}{25}+\frac{41932 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{3875 \sqrt{31}} \]

[Out]

(4*x)/25 + (121*(61 + 69*x))/(3875*(2 + 3*x + 5*x^2)) + (41932*ArcTan[(3 + 10*x)
/Sqrt[31]])/(3875*Sqrt[31]) - (22*Log[2 + 3*x + 5*x^2])/125

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Rubi [A]  time = 0.110633, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}-\frac{22}{125} \log \left (5 x^2+3 x+2\right )+\frac{4 x}{25}+\frac{41932 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{3875 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x)/25 + (121*(61 + 69*x))/(3875*(2 + 3*x + 5*x^2)) + (41932*ArcTan[(3 + 10*x)
/Sqrt[31]])/(3875*Sqrt[31]) - (22*Log[2 + 3*x + 5*x^2])/125

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{8 x^{3}}{31} + \frac{\left (10 x + 3\right ) \left (2 x^{2} - x + 3\right )^{2}}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{22 \log{\left (5 x^{2} + 3 x + 2 \right )}}{125} + \frac{41932 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{120125} - \frac{\int \frac{542}{25}\, dx}{31} + \frac{104 \int x\, dx}{155} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*x**3/31 + (10*x + 3)*(2*x**2 - x + 3)**2/(31*(5*x**2 + 3*x + 2)) - 22*log(5*x
**2 + 3*x + 2)/125 + 41932*sqrt(31)*atan(sqrt(31)*(10*x/31 + 3/31))/120125 - Int
egral(542/25, x)/31 + 104*Integral(x, x)/155

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Mathematica [A]  time = 0.0554694, size = 59, normalized size = 0.94 \[ \frac{\frac{3751 (69 x+61)}{5 x^2+3 x+2}-21142 \log \left (5 x^2+3 x+2\right )+19220 x+41932 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{120125} \]

Antiderivative was successfully verified.

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

[Out]

(19220*x + (3751*(61 + 69*x))/(2 + 3*x + 5*x^2) + 41932*Sqrt[31]*ArcTan[(3 + 10*
x)/Sqrt[31]] - 21142*Log[2 + 3*x + 5*x^2])/120125

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Maple [A]  time = 0.009, size = 51, normalized size = 0.8 \[{\frac{4\,x}{25}}-{\frac{11}{25} \left ( -{\frac{759\,x}{775}}-{\frac{671}{775}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}-{\frac{22\,\ln \left ( 25\,{x}^{2}+15\,x+10 \right ) }{125}}+{\frac{41932\,\sqrt{31}}{120125}\arctan \left ({\frac{ \left ( 50\,x+15 \right ) \sqrt{31}}{155}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/25*x-11/25*(-759/775*x-671/775)/(x^2+3/5*x+2/5)-22/125*ln(25*x^2+15*x+10)+4193
2/120125*31^(1/2)*arctan(1/155*(50*x+15)*31^(1/2))

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Maxima [A]  time = 0.775687, size = 70, normalized size = 1.11 \[ \frac{41932}{120125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{4}{25} \, x + \frac{121 \,{\left (69 \, x + 61\right )}}{3875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{22}{125} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

41932/120125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 4/25*x + 121/3875*(69*x
+ 61)/(5*x^2 + 3*x + 2) - 22/125*log(5*x^2 + 3*x + 2)

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Fricas [A]  time = 0.259865, size = 117, normalized size = 1.86 \[ -\frac{\sqrt{31}{\left (682 \, \sqrt{31}{\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 41932 \,{\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \sqrt{31}{\left (3100 \, x^{3} + 1860 \, x^{2} + 9589 \, x + 7381\right )}\right )}}{120125 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/120125*sqrt(31)*(682*sqrt(31)*(5*x^2 + 3*x + 2)*log(5*x^2 + 3*x + 2) - 41932*
(5*x^2 + 3*x + 2)*arctan(1/31*sqrt(31)*(10*x + 3)) - sqrt(31)*(3100*x^3 + 1860*x
^2 + 9589*x + 7381))/(5*x^2 + 3*x + 2)

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Sympy [A]  time = 0.209593, size = 65, normalized size = 1.03 \[ \frac{4 x}{25} + \frac{8349 x + 7381}{19375 x^{2} + 11625 x + 7750} - \frac{22 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{125} + \frac{41932 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{120125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x/25 + (8349*x + 7381)/(19375*x**2 + 11625*x + 7750) - 22*log(x**2 + 3*x/5 + 2
/5)/125 + 41932*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/120125

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GIAC/XCAS [A]  time = 0.264785, size = 70, normalized size = 1.11 \[ \frac{41932}{120125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{4}{25} \, x + \frac{121 \,{\left (69 \, x + 61\right )}}{3875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{22}{125} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

41932/120125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 4/25*x + 121/3875*(69*x
+ 61)/(5*x^2 + 3*x + 2) - 22/125*ln(5*x^2 + 3*x + 2)