3.1241 \(\int \frac{c+d x}{a+b x} \, dx\)

Optimal. Leaf size=25 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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Rubi [A]  time = 0.0407364, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a + b*x),x]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0121552, size = 25, normalized size = 1. \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

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

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

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Maple [A]  time = 0.004, size = 32, normalized size = 1.3 \[{\frac{dx}{b}}-{\frac{a\ln \left ( bx+a \right ) d}{{b}^{2}}}+{\frac{c\ln \left ( bx+a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(b*x+a),x)

[Out]

d*x/b-1/b^2*ln(b*x+a)*a*d+c*ln(b*x+a)/b

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Maxima [A]  time = 1.34197, size = 34, normalized size = 1.36 \[ \frac{d x}{b} + \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/b + (b*c - a*d)*log(b*x + a)/b^2

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Fricas [A]  time = 0.196054, size = 32, normalized size = 1.28 \[ \frac{b d x +{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*d*x + (b*c - a*d)*log(b*x + a))/b^2

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Sympy [A]  time = 1.13274, size = 20, normalized size = 0.8 \[ \frac{d x}{b} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(b*x+a),x)

[Out]

d*x/b - (a*d - b*c)*log(a + b*x)/b**2

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GIAC/XCAS [A]  time = 0.219165, size = 35, normalized size = 1.4 \[ \frac{d x}{b} + \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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