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

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 = 0.0184414, 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, Rules used = {72} \[ \frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b} \]

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)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{A+B x}{x (a+b x)} \, dx &=\int \left (\frac{A}{a x}+\frac{-A b+a B}{a (a+b x)}\right ) \, dx\\ &=\frac{A \log (x)}{a}-\frac{(A b-a B) \log (a+b x)}{a b}\\ \end{align*}

Mathematica [A]  time = 0.0097053, 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 = 0.005, size = 32, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( bx+a \right ) A}{a}}+{\frac{\ln \left ( bx+a \right ) B}{b}} \end{align*}

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 = 1.01215, size = 39, normalized size = 1.3 \begin{align*} \frac{A \log \left (x\right )}{a} + \frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x/(b*x+a),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 = 1.62748, size = 63, normalized size = 2.1 \begin{align*} \frac{A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x/(b*x+a),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 = 0.528548, size = 41, normalized size = 1.37 \begin{align*} \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} \end{align*}

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 [A]  time = 1.24199, size = 42, normalized size = 1.4 \begin{align*} \frac{A \log \left ({\left | x \right |}\right )}{a} + \frac{{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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