3.88 \(\int \frac{1}{a+c x+b x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]

[Out]

(2*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/Sqrt[4*a*b - c^2]

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Rubi [A]  time = 0.0649527, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x + b*x^2)^(-1),x]

[Out]

(2*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/Sqrt[4*a*b - c^2]

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Rubi in Sympy [A]  time = 4.59494, size = 34, normalized size = 0.89 \[ - \frac{2 \operatorname{atanh}{\left (\frac{2 b x + c}{\sqrt{- 4 a b + c^{2}}} \right )}}{\sqrt{- 4 a b + c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+c*x+a),x)

[Out]

-2*atanh((2*b*x + c)/sqrt(-4*a*b + c**2))/sqrt(-4*a*b + c**2)

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Mathematica [A]  time = 0.0190511, size = 38, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 b x+c}{\sqrt{4 a b-c^2}}\right )}{\sqrt{4 a b-c^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x + b*x^2)^(-1),x]

[Out]

(2*ArcTan[(c + 2*b*x)/Sqrt[4*a*b - c^2]])/Sqrt[4*a*b - c^2]

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ 2\,{\frac{1}{\sqrt{4\,ab-{c}^{2}}}\arctan \left ({\frac{2\,bx+c}{\sqrt{4\,ab-{c}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+c*x+a),x)

[Out]

2*arctan((2*b*x+c)/(4*a*b-c^2)^(1/2))/(4*a*b-c^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^2 + c*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217455, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{4 \, a b c - c^{3} + 2 \,{\left (4 \, a b^{2} - b c^{2}\right )} x +{\left (2 \, b^{2} x^{2} + 2 \, b c x - 2 \, a b + c^{2}\right )} \sqrt{-4 \, a b + c^{2}}}{b x^{2} + c x + a}\right )}{\sqrt{-4 \, a b + c^{2}}}, -\frac{2 \, \arctan \left (-\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{\sqrt{4 \, a b - c^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^2 + c*x + a),x, algorithm="fricas")

[Out]

[log((4*a*b*c - c^3 + 2*(4*a*b^2 - b*c^2)*x + (2*b^2*x^2 + 2*b*c*x - 2*a*b + c^2
)*sqrt(-4*a*b + c^2))/(b*x^2 + c*x + a))/sqrt(-4*a*b + c^2), -2*arctan(-(2*b*x +
c)/sqrt(4*a*b - c^2))/sqrt(4*a*b - c^2)]

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Sympy [A]  time = 0.552638, size = 124, normalized size = 3.26 \[ - \sqrt{- \frac{1}{4 a b - c^{2}}} \log{\left (x + \frac{- 4 a b \sqrt{- \frac{1}{4 a b - c^{2}}} + c^{2} \sqrt{- \frac{1}{4 a b - c^{2}}} + c}{2 b} \right )} + \sqrt{- \frac{1}{4 a b - c^{2}}} \log{\left (x + \frac{4 a b \sqrt{- \frac{1}{4 a b - c^{2}}} - c^{2} \sqrt{- \frac{1}{4 a b - c^{2}}} + c}{2 b} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+c*x+a),x)

[Out]

-sqrt(-1/(4*a*b - c**2))*log(x + (-4*a*b*sqrt(-1/(4*a*b - c**2)) + c**2*sqrt(-1/
(4*a*b - c**2)) + c)/(2*b)) + sqrt(-1/(4*a*b - c**2))*log(x + (4*a*b*sqrt(-1/(4*
a*b - c**2)) - c**2*sqrt(-1/(4*a*b - c**2)) + c)/(2*b))

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GIAC/XCAS [A]  time = 0.20853, size = 46, normalized size = 1.21 \[ \frac{2 \, \arctan \left (\frac{2 \, b x + c}{\sqrt{4 \, a b - c^{2}}}\right )}{\sqrt{4 \, a b - c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^2 + c*x + a),x, algorithm="giac")

[Out]

2*arctan((2*b*x + c)/sqrt(4*a*b - c^2))/sqrt(4*a*b - c^2)