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

Optimal. Leaf size=13 \[ \log \left (a-b x-c x^2\right ) \]

[Out]

Log[a - b*x - c*x^2]

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Rubi [A]  time = 0.00850707, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \log \left (a-b x-c x^2\right ) \]

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 = 4.01225, size = 10, normalized size = 0.77 \[ \log{\left (- a + b x + c x^{2} \right )} \]

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 = 0.00761912, size = 12, normalized size = 0.92 \[ \log (x (b+c x)-a) \]

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 = 14, 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 = 0.750719, size = 18, normalized size = 1.38 \[ \log \left (c x^{2} + b x - a\right ) \]

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 = 0.274267, size = 18, normalized size = 1.38 \[ \log \left (c x^{2} + b x - a\right ) \]

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 = 1.18518, size = 10, normalized size = 0.77 \[ \log{\left (- a + b x + c x^{2} \right )} \]

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 = 0.26508, size = 18, normalized size = 1.38 \[{\rm ln}\left (c x^{2} + b x - a\right ) \]

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)