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

Optimal. Leaf size=64 \[ \frac{121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}+\frac{11 (45710 x+17557)}{240250 \left (5 x^2+3 x+2\right )}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]

[Out]

(121*(61 + 69*x))/(7750*(2 + 3*x + 5*x^2)^2) + (11*(17557 + 45710*x))/(240250*(2
+ 3*x + 5*x^2)) + (4330*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])

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Rubi [A]  time = 0.0937121, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}+\frac{11 (45710 x+17557)}{240250 \left (5 x^2+3 x+2\right )}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(121*(61 + 69*x))/(7750*(2 + 3*x + 5*x^2)^2) + (11*(17557 + 45710*x))/(240250*(2
+ 3*x + 5*x^2)) + (4330*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (10 x + 3\right ) \left (2 x^{2} - x + 3\right )^{2}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{22 \left (1712 x + 1053\right )}{24025 \left (5 x^{2} + 3 x + 2\right )} + \frac{4330 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{29791} - \frac{\int \frac{8}{5}\, dx}{62} \]

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)**3,x)

[Out]

(10*x + 3)*(2*x**2 - x + 3)**2/(62*(5*x**2 + 3*x + 2)**2) + 22*(1712*x + 1053)/(
24025*(5*x**2 + 3*x + 2)) + 4330*sqrt(31)*atan(sqrt(31)*(10*x/31 + 3/31))/29791
- Integral(8/5, x)/62

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Mathematica [A]  time = 0.0513349, size = 53, normalized size = 0.83 \[ \frac{11 \left (45710 x^3+44983 x^2+33524 x+11183\right )}{48050 \left (5 x^2+3 x+2\right )^2}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(11*(11183 + 33524*x + 44983*x^2 + 45710*x^3))/(48050*(2 + 3*x + 5*x^2)^2) + (43
30*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])

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Maple [A]  time = 0.008, size = 47, normalized size = 0.7 \[ 25\,{\frac{1}{ \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{50281\,{x}^{3}}{120125}}+{\frac{494813\,{x}^{2}}{1201250}}+{\frac{184382\,x}{600625}}+{\frac{123013}{1201250}} \right ) }+{\frac{4330\,\sqrt{31}}{29791}\arctan \left ({\frac{ \left ( 250\,x+75 \right ) \sqrt{31}}{775}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(50281/120125*x^3+494813/1201250*x^2+184382/600625*x+123013/1201250)/(5*x^2+3
*x+2)^2+4330/29791*31^(1/2)*arctan(1/775*(250*x+75)*31^(1/2))

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Maxima [A]  time = 0.771819, size = 76, normalized size = 1.19 \[ \frac{4330}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4330/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 11/48050*(45710*x^3 + 449
83*x^2 + 33524*x + 11183)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Fricas [A]  time = 0.258176, size = 109, normalized size = 1.7 \[ \frac{\sqrt{31}{\left (216500 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 11 \, \sqrt{31}{\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}\right )}}{1489550 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1489550*sqrt(31)*(216500*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqr
t(31)*(10*x + 3)) + 11*sqrt(31)*(45710*x^3 + 44983*x^2 + 33524*x + 11183))/(25*x
^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Sympy [A]  time = 0.2428, size = 63, normalized size = 0.98 \[ \frac{502810 x^{3} + 494813 x^{2} + 368764 x + 123013}{1201250 x^{4} + 1441500 x^{3} + 1393450 x^{2} + 576600 x + 192200} + \frac{4330 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{29791} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(502810*x**3 + 494813*x**2 + 368764*x + 123013)/(1201250*x**4 + 1441500*x**3 + 1
393450*x**2 + 576600*x + 192200) + 4330*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(
31)/31)/29791

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GIAC/XCAS [A]  time = 0.265557, size = 62, normalized size = 0.97 \[ \frac{4330}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4330/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 11/48050*(45710*x^3 + 449
83*x^2 + 33524*x + 11183)/(5*x^2 + 3*x + 2)^2