3.1058 \(\int \frac{a c-b c x}{(a+b x)^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0132458, antiderivative size = 27, 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} \[ -\frac{2 a c}{b (a+b x)}-\frac{c \log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)/(a + b*x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.009045, size = 23, normalized size = 0.85 \[ -\frac{c \left (\frac{2 a}{a+b x}+\log (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 28, normalized size = 1. \begin{align*} -2\,{\frac{ac}{b \left ( bx+a \right ) }}-{\frac{c\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02968, size = 38, normalized size = 1.41 \begin{align*} -\frac{2 \, a c}{b^{2} x + a b} - \frac{c \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.53493, size = 72, normalized size = 2.67 \begin{align*} -\frac{2 \, a c +{\left (b c x + a c\right )} \log \left (b x + a\right )}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.318086, size = 24, normalized size = 0.89 \begin{align*} - \frac{2 a c}{a b + b^{2} x} - \frac{c \log{\left (a + b x \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.05256, size = 73, normalized size = 2.7 \begin{align*} c{\left (\frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )} - \frac{a c}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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