3.1033 \(\int \frac{a+b x}{(a c-b c x)^2} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2} \]

[Out]

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

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Rubi [A]  time = 0.017297, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^2,x]

[Out]

(2*a)/(b*c^2*(a - b*x)) + Log[a - b*x]/(b*c^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 c-b c x)^2} \, dx &=\int \left (\frac{2 a}{c^2 (a-b x)^2}-\frac{1}{c^2 (a-b x)}\right ) \, dx\\ &=\frac{2 a}{b c^2 (a-b x)}+\frac{\log (a-b x)}{b c^2}\\ \end{align*}

Mathematica [A]  time = 0.0166424, size = 28, normalized size = 0.88 \[ \frac{\log (c (a-b x))+\frac{2 a}{a-b x}}{b c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 35, normalized size = 1.1 \begin{align*} -2\,{\frac{a}{{c}^{2}b \left ( bx-a \right ) }}+{\frac{\ln \left ( bx-a \right ) }{{c}^{2}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b*c*x+a*c)^2,x)

[Out]

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

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Maxima [A]  time = 1.02122, size = 50, normalized size = 1.56 \begin{align*} -\frac{2 \, a}{b^{2} c^{2} x - a b c^{2}} + \frac{\log \left (b x - a\right )}{b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.41035, size = 73, normalized size = 2.28 \begin{align*} \frac{{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} c^{2} x - a b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.338311, size = 29, normalized size = 0.91 \begin{align*} - \frac{2 a}{- a b c^{2} + b^{2} c^{2} x} + \frac{\log{\left (- a + b x \right )}}{b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)**2,x)

[Out]

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

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Giac [B]  time = 1.07027, size = 109, normalized size = 3.41 \begin{align*} -\frac{\frac{a}{{\left (b c x - a c\right )} b} + \frac{\log \left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c}}{c} - \frac{a}{{\left (b c x - a c\right )} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^2,x, algorithm="giac")

[Out]

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