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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.91, size = 28, normalized size = 0.93 \[ \frac {A b \log \relax (x) + {\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/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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giac [A]  time = 1.09, size = 31, normalized size = 1.03 \[ \frac {A \log \left ({\left | x \right |}\right )}{a} + \frac {{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \]

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)

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maple [A]  time = 0.00, size = 32, normalized size = 1.07 \[ \frac {A \ln \relax (x )}{a}-\frac {A \ln \left (b x +a \right )}{a}+\frac {B \ln \left (b x +a \right )}{b} \]

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 = 1.14, size = 29, normalized size = 0.97 \[ \frac {A \log \relax (x)}{a} + \frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/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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mupad [B]  time = 0.12, size = 28, normalized size = 0.93 \[ \frac {A\,\ln \relax (x)}{a}-\ln \left (a+b\,x\right )\,\left (\frac {A}{a}-\frac {B}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x*(a + b*x)),x)

[Out]

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

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sympy [A]  time = 0.65, size = 41, normalized size = 1.37 \[ \frac {A \log {\relax (x )}}{a} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a + \frac {a \left (- A b + B a\right )}{b}}{- 2 A b + B a} \right )}}{a b} \]

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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