3.1931 \(\int \frac{a+b x}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=10 \[ \frac{\log (a+b x)}{b} \]

[Out]

Log[a + b*x]/b

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Rubi [A]  time = 0.00822708, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\log (a+b x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x]/b

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Rubi in Sympy [A]  time = 10.3111, size = 7, normalized size = 0.7 \[ \frac{\log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x)/b

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Mathematica [A]  time = 0.00177559, size = 10, normalized size = 1. \[ \frac{\log (a+b x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x]/b

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \[{\frac{\ln \left ( bx+a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(b*x+a)/b

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Maxima [A]  time = 0.706625, size = 30, normalized size = 3. \[ \frac{\log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(b^2*x^2 + 2*a*b*x + a^2)/b

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Fricas [A]  time = 0.269749, size = 14, normalized size = 1.4 \[ \frac{\log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x + a)/b

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Sympy [A]  time = 0.11106, size = 7, normalized size = 0.7 \[ \frac{\log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x)/b

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GIAC/XCAS [A]  time = 0.283294, size = 15, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x + a))/b