3.161 \(\int \frac{x}{a+b x} \, dx\)

Optimal. Leaf size=18 \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

[Out]

x/b - (a*Log[a + b*x])/b^2

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Rubi [A]  time = 0.0225335, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x),x]

[Out]

x/b - (a*Log[a + b*x])/b^2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b x \right )}}{b^{2}} + \int \frac{1}{b}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x+a),x)

[Out]

-a*log(a + b*x)/b**2 + Integral(1/b, x)

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Mathematica [A]  time = 0.00347118, size = 18, normalized size = 1. \[ \frac{x}{b}-\frac{a \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x),x]

[Out]

x/b - (a*Log[a + b*x])/b^2

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Maple [A]  time = 0.004, size = 19, normalized size = 1.1 \[{\frac{x}{b}}-{\frac{a\ln \left ( bx+a \right ) }{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a),x)

[Out]

x/b-a*ln(b*x+a)/b^2

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Maxima [A]  time = 1.34242, size = 24, normalized size = 1.33 \[ \frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/b - a*log(b*x + a)/b^2

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Fricas [A]  time = 0.191706, size = 23, normalized size = 1.28 \[ \frac{b x - a \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x - a*log(b*x + a))/b^2

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Sympy [A]  time = 1.02025, size = 14, normalized size = 0.78 \[ - \frac{a \log{\left (a + b x \right )}}{b^{2}} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a),x)

[Out]

-a*log(a + b*x)/b**2 + x/b

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GIAC/XCAS [A]  time = 0.2246, size = 26, normalized size = 1.44 \[ \frac{x}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/b - a*ln(abs(b*x + a))/b^2