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

Optimal. Leaf size=11 \[ \log \left (a+b x+c x^2\right ) \]

[Out]

Log[a + b*x + c*x^2]

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Rubi [A]  time = 0.00832596, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \log \left (a+b x+c x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)/(a + b*x + c*x^2),x]

[Out]

Log[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 3.69863, size = 10, normalized size = 0.91 \[ \log{\left (a + b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x + c*x**2)

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Mathematica [A]  time = 0.00455208, size = 10, normalized size = 0.91 \[ \log (a+x (b+c x)) \]

Antiderivative was successfully verified.

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

[Out]

Log[a + x*(b + c*x)]

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Maple [A]  time = 0.002, size = 12, normalized size = 1.1 \[ \ln \left ( c{x}^{2}+bx+a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x+a),x)

[Out]

ln(c*x^2+b*x+a)

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Maxima [A]  time = 0.772864, size = 15, normalized size = 1.36 \[ \log \left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^2 + b*x + a)

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Fricas [A]  time = 0.2787, size = 15, normalized size = 1.36 \[ \log \left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^2 + b*x + a)

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Sympy [A]  time = 1.16436, size = 10, normalized size = 0.91 \[ \log{\left (a + b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.264829, size = 15, normalized size = 1.36 \[{\rm ln}\left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(c*x^2 + b*x + a)