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

Optimal. Leaf size=64 \[ \frac{553 (10 x+3)}{9610 \left (5 x^2+3 x+2\right )}+\frac{11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}+\frac{1106 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]

[Out]

(11*(7 + 13*x))/(310*(2 + 3*x + 5*x^2)^2) + (553*(3 + 10*x))/(9610*(2 + 3*x + 5*
x^2)) + (1106*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])

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Rubi [A]  time = 0.0659882, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{553 (10 x+3)}{9610 \left (5 x^2+3 x+2\right )}+\frac{11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}+\frac{1106 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(11*(7 + 13*x))/(310*(2 + 3*x + 5*x^2)^2) + (553*(3 + 10*x))/(9610*(2 + 3*x + 5*
x^2)) + (1106*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])

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Rubi in Sympy [A]  time = 9.85074, size = 56, normalized size = 0.88 \[ \frac{553 \left (10 x + 3\right )}{9610 \left (5 x^{2} + 3 x + 2\right )} + \frac{143 x + 77}{310 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{1106 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{29791} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

553*(10*x + 3)/(9610*(5*x**2 + 3*x + 2)) + (143*x + 77)/(310*(5*x**2 + 3*x + 2)*
*2) + 1106*sqrt(31)*atan(sqrt(31)*(10*x/31 + 3/31))/29791

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Mathematica [A]  time = 0.0518193, size = 53, normalized size = 0.83 \[ \frac{\frac{31 \left (5530 x^3+4977 x^2+4094 x+1141\right )}{\left (5 x^2+3 x+2\right )^2}+2212 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{59582} \]

Antiderivative was successfully verified.

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

[Out]

((31*(1141 + 4094*x + 4977*x^2 + 5530*x^3))/(2 + 3*x + 5*x^2)^2 + 2212*Sqrt[31]*
ArcTan[(3 + 10*x)/Sqrt[31]])/59582

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Maple [A]  time = 0.009, size = 47, normalized size = 0.7 \[ 25\,{\frac{1}{ \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{553\,{x}^{3}}{4805}}+{\frac{4977\,{x}^{2}}{48050}}+{\frac{2047\,x}{24025}}+{\frac{1141}{48050}} \right ) }+{\frac{1106\,\sqrt{31}}{29791}\arctan \left ({\frac{ \left ( 250\,x+75 \right ) \sqrt{31}}{775}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(553/4805*x^3+4977/48050*x^2+2047/24025*x+1141/48050)/(5*x^2+3*x+2)^2+1106/29
791*31^(1/2)*arctan(1/775*(250*x+75)*31^(1/2))

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Maxima [A]  time = 0.786788, size = 76, normalized size = 1.19 \[ \frac{1106}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{5530 \, x^{3} + 4977 \, x^{2} + 4094 \, x + 1141}{1922 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1106/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/1922*(5530*x^3 + 4977*x
^2 + 4094*x + 1141)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Fricas [A]  time = 0.267808, size = 108, normalized size = 1.69 \[ \frac{\sqrt{31}{\left (2212 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \sqrt{31}{\left (5530 \, x^{3} + 4977 \, x^{2} + 4094 \, x + 1141\right )}\right )}}{59582 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/59582*sqrt(31)*(2212*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqrt(31
)*(10*x + 3)) + sqrt(31)*(5530*x^3 + 4977*x^2 + 4094*x + 1141))/(25*x^4 + 30*x^3
+ 29*x^2 + 12*x + 4)

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Sympy [A]  time = 0.225381, size = 63, normalized size = 0.98 \[ \frac{5530 x^{3} + 4977 x^{2} + 4094 x + 1141}{48050 x^{4} + 57660 x^{3} + 55738 x^{2} + 23064 x + 7688} + \frac{1106 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{29791} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5530*x**3 + 4977*x**2 + 4094*x + 1141)/(48050*x**4 + 57660*x**3 + 55738*x**2 +
23064*x + 7688) + 1106*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/29791

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GIAC/XCAS [A]  time = 0.264558, size = 62, normalized size = 0.97 \[ \frac{1106}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{5530 \, x^{3} + 4977 \, x^{2} + 4094 \, x + 1141}{1922 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1106/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/1922*(5530*x^3 + 4977*x
^2 + 4094*x + 1141)/(5*x^2 + 3*x + 2)^2