3.148 \(\int \frac{A+B x}{x (a+b x)} \, dx\)

Optimal. Leaf size=30 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b} \]

[Out]

(A*Log[x])/a - ((A*b - a*B)*Log[a + b*x])/(a*b)

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Rubi [A]  time = 0.0515285, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x*(a + b*x)),x]

[Out]

(A*Log[x])/a - ((A*b - a*B)*Log[a + b*x])/(a*b)

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Rubi in Sympy [A]  time = 12.0976, size = 22, normalized size = 0.73 \[ \frac{A \log{\left (x \right )}}{a} - \frac{\left (A b - B a\right ) \log{\left (a + b x \right )}}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/a - (A*b - B*a)*log(a + b*x)/(a*b)

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Mathematica [A]  time = 0.0157441, size = 29, normalized size = 0.97 \[ \frac{(a B-A b) \log (a+b x)}{a b}+\frac{A \log (x)}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x*(a + b*x)),x]

[Out]

(A*Log[x])/a + ((-(A*b) + a*B)*Log[a + b*x])/(a*b)

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Maple [A]  time = 0.008, size = 32, normalized size = 1.1 \[{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( bx+a \right ) A}{a}}+{\frac{\ln \left ( bx+a \right ) B}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x/(b*x+a),x)

[Out]

A*ln(x)/a-1/a*ln(b*x+a)*A+1/b*ln(b*x+a)*B

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Maxima [A]  time = 1.37698, size = 39, normalized size = 1.3 \[ \frac{A \log \left (x\right )}{a} + \frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*log(x)/a + (B*a - A*b)*log(b*x + a)/(a*b)

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Fricas [A]  time = 0.209175, size = 38, normalized size = 1.27 \[ \frac{A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b*log(x) + (B*a - A*b)*log(b*x + a))/(a*b)

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Sympy [A]  time = 2.01345, size = 41, normalized size = 1.37 \[ \frac{A \log{\left (x \right )}}{a} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a + \frac{a \left (- A b + B a\right )}{b}}{- 2 A b + B a} \right )}}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x/(b*x+a),x)

[Out]

A*log(x)/a + (-A*b + B*a)*log(x + (-A*a + a*(-A*b + B*a)/b)/(-2*A*b + B*a))/(a*b
)

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GIAC/XCAS [A]  time = 0.3275, size = 42, normalized size = 1.4 \[ \frac{A{\rm ln}\left ({\left | x \right |}\right )}{a} + \frac{{\left (B a - A b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*ln(abs(x))/a + (B*a - A*b)*ln(abs(b*x + a))/(a*b)