3.173 \(\int \frac{x}{(a+b x)^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

[Out]

a/(b^2*(a + b*x)) + Log[a + b*x]/b^2

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Rubi [A]  time = 0.0119928, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a}{b^2 (a+b x)}+\frac{\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0105572, size = 20, normalized size = 0.87 \[ \frac{\frac{a}{a+b x}+\log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^2,x]

[Out]

(a/(a + b*x) + Log[a + b*x])/b^2

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Maple [A]  time = 0.004, size = 24, normalized size = 1. \begin{align*}{\frac{a}{{b}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/b^2/(b*x+a)+ln(b*x+a)/b^2

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Maxima [A]  time = 1.02621, size = 35, normalized size = 1.52 \begin{align*} \frac{a}{b^{3} x + a b^{2}} + \frac{\log \left (b x + a\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(b^3*x + a*b^2) + log(b*x + a)/b^2

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Fricas [A]  time = 1.50717, size = 62, normalized size = 2.7 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.399283, size = 20, normalized size = 0.87 \begin{align*} \frac{a}{a b^{2} + b^{3} x} + \frac{\log{\left (a + b x \right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(a*b**2 + b**3*x) + log(a + b*x)/b**2

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Giac [A]  time = 1.17541, size = 57, normalized size = 2.48 \begin{align*} -\frac{\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}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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