\(\int \frac {c+d x}{(a+b x)^2} \, dx\) [1242]

Optimal result

Integrand size = 13, antiderivative size = 32 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=-\frac {b c-a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {c+d x}{(a+b x)^2} \, dx=\frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \]

[In]

Int[(c + d*x)/(a + b*x)^2,x]

[Out]

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

Rule 45

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=\frac {-b c+a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \]

[In]

Integrate[(c + d*x)/(a + b*x)^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {a d -b c}{b^{2} \left (b x +a \right )}+\frac {d \ln \left (b x +a \right )}{b^{2}}\)	\(32\)
default	\(\frac {d \ln \left (b x +a \right )}{b^{2}}-\frac {-a d +b c}{b^{2} \left (b x +a \right )}\)	\(33\)
risch	\(\frac {a d}{b^{2} \left (b x +a \right )}-\frac {c}{b \left (b x +a \right )}+\frac {d \ln \left (b x +a \right )}{b^{2}}\)	\(39\)
parallelrisch	\(\frac {\ln \left (b x +a \right ) x b d +\ln \left (b x +a \right ) a d +a d -b c}{b^{2} \left (b x +a \right )}\)	\(39\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=-\frac {b c - a d - {\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=\frac {a d - b c}{a b^{2} + b^{3} x} + \frac {d \log {\left (a + b x \right )}}{b^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=-\frac {b c - a d}{b^{3} x + a b^{2}} + \frac {d \log \left (b x + a\right )}{b^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=-\frac {d {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}\right )}}{b} - \frac {c}{{\left (b x + a\right )} b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x}{(a+b x)^2} \, dx=\frac {a\,d-b\,c}{b^2\,\left (a+b\,x\right )}+\frac {d\,\ln \left (a+b\,x\right )}{b^2} \]

[In]

int((c + d*x)/(a + b*x)^2,x)

[Out]

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