3.1347 \(\int \frac{a+b x}{(c+d x)^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0208401, antiderivative size = 31, 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} \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c + d*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0101126, size = 31, normalized size = 1. \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(c + d*x)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 39, normalized size = 1.3 \begin{align*} -{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }}+{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.943292, size = 46, normalized size = 1.48 \begin{align*} \frac{b c - a d}{d^{3} x + c d^{2}} + \frac{b \log \left (d x + c\right )}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c - a*d)/(d^3*x + c*d^2) + b*log(d*x + c)/d^2

________________________________________________________________________________________

Fricas [A]  time = 1.62261, size = 78, normalized size = 2.52 \begin{align*} \frac{b c - a d +{\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c - a*d + (b*d*x + b*c)*log(d*x + c))/(d^3*x + c*d^2)

________________________________________________________________________________________

Sympy [A]  time = 0.336933, size = 27, normalized size = 0.87 \begin{align*} \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{c d^{2} + d^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(c + d*x)/d**2 - (a*d - b*c)/(c*d**2 + d**3*x)

________________________________________________________________________________________

Giac [A]  time = 1.0497, size = 77, normalized size = 2.48 \begin{align*} -\frac{b{\left (\frac{\log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d} - \frac{c}{{\left (d x + c\right )} d}\right )}}{d} - \frac{a}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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