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

Optimal. Leaf size=42 \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]

[Out]

(2*x)/5 + (143*ArcTan[(3 + 10*x)/Sqrt[31]])/(25*Sqrt[31]) - (11*Log[2 + 3*x + 5*
x^2])/50

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Rubi [A]  time = 0.079735, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x)/5 + (143*ArcTan[(3 + 10*x)/Sqrt[31]])/(25*Sqrt[31]) - (11*Log[2 + 3*x + 5*
x^2])/50

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{11 \log{\left (5 x^{2} + 3 x + 2 \right )}}{50} + \frac{143 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{775} + \int \frac{2}{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*log(5*x**2 + 3*x + 2)/50 + 143*sqrt(31)*atan(sqrt(31)*(10*x/31 + 3/31))/775
+ Integral(2/5, x)

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Mathematica [A]  time = 0.0278974, size = 42, normalized size = 1. \[ -\frac{11}{50} \log \left (5 x^2+3 x+2\right )+\frac{2 x}{5}+\frac{143 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{25 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x)/5 + (143*ArcTan[(3 + 10*x)/Sqrt[31]])/(25*Sqrt[31]) - (11*Log[2 + 3*x + 5*
x^2])/50

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Maple [A]  time = 0.006, size = 34, normalized size = 0.8 \[{\frac{2\,x}{5}}-{\frac{11\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{50}}+{\frac{143\,\sqrt{31}}{775}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x-11/50*ln(5*x^2+3*x+2)+143/775*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Maxima [A]  time = 0.782048, size = 45, normalized size = 1.07 \[ \frac{143}{775} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{2}{5} \, x - \frac{11}{50} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

143/775*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 2/5*x - 11/50*log(5*x^2 + 3*
x + 2)

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Fricas [A]  time = 0.261923, size = 55, normalized size = 1.31 \[ \frac{1}{1550} \, \sqrt{31}{\left (20 \, \sqrt{31} x - 11 \, \sqrt{31} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 286 \, \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1550*sqrt(31)*(20*sqrt(31)*x - 11*sqrt(31)*log(5*x^2 + 3*x + 2) + 286*arctan(1
/31*sqrt(31)*(10*x + 3)))

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Sympy [A]  time = 0.134828, size = 49, normalized size = 1.17 \[ \frac{2 x}{5} - \frac{11 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{50} + \frac{143 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{775} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x/5 - 11*log(x**2 + 3*x/5 + 2/5)/50 + 143*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*s
qrt(31)/31)/775

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GIAC/XCAS [A]  time = 0.265926, size = 45, normalized size = 1.07 \[ \frac{143}{775} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{2}{5} \, x - \frac{11}{50} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

143/775*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 2/5*x - 11/50*ln(5*x^2 + 3*x
+ 2)