3.1116 \(\int \frac{A+B x}{a+b x} \, dx\)

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

[Out]

(B*x)/b + ((A*b - a*B)*Log[a + b*x])/b^2

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Rubi [A]  time = 0.0179, 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, Rules used = {43} \[ \frac{(A b-a B) \log (a+b x)}{b^2}+\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(a + b*x),x]

[Out]

(B*x)/b + ((A*b - a*B)*Log[a + b*x])/b^2

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{A+B x}{a+b x} \, dx &=\int \left (\frac{B}{b}+\frac{A b-a B}{b (a+b x)}\right ) \, dx\\ &=\frac{B x}{b}+\frac{(A b-a B) \log (a+b x)}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.0070388, size = 25, normalized size = 1. \[ \frac{(A b-a B) \log (a+b x)}{b^2}+\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(a + b*x),x]

[Out]

(B*x)/b + ((A*b - a*B)*Log[a + b*x])/b^2

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Maple [A]  time = 0.002, size = 32, normalized size = 1.3 \begin{align*}{\frac{Bx}{b}}+{\frac{\ln \left ( bx+a \right ) A}{b}}-{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a),x)

[Out]

B*x/b+1/b*ln(b*x+a)*A-1/b^2*ln(b*x+a)*B*a

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Maxima [A]  time = 1.24185, size = 35, normalized size = 1.4 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b - (B*a - A*b)*log(b*x + a)/b^2

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Fricas [A]  time = 1.766, size = 54, normalized size = 2.16 \begin{align*} \frac{B b x -{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.288708, size = 20, normalized size = 0.8 \begin{align*} \frac{B x}{b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.80959, size = 36, normalized size = 1.44 \begin{align*} \frac{B x}{b} - \frac{{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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