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

Optimal. Leaf size=29 \[ \frac{x \log (x)}{a (a+b x)}-\frac{\log (a+b x)}{a b} \]

[Out]

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

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Rubi [A]  time = 0.013455, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2314, 31} \[ \frac{x \log (x)}{a (a+b x)}-\frac{\log (a+b x)}{a b} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(a + b*x)^2,x]

[Out]

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

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0164849, size = 27, normalized size = 0.93 \[ \frac{\frac{x \log (x)}{a+b x}-\frac{\log (a+b x)}{b}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 30, normalized size = 1. \begin{align*}{\frac{x\ln \left ( x \right ) }{a \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a \right ) }{ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.93178, size = 51, normalized size = 1.76 \begin{align*} -\frac{\frac{\log \left (b x + a\right )}{a} - \frac{\log \left (x\right )}{a}}{b} - \frac{\log \left (x\right )}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.22414, size = 77, normalized size = 2.66 \begin{align*} \frac{b x \log \left (x\right ) -{\left (b x + a\right )} \log \left (b x + a\right )}{a b^{2} x + a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.358122, size = 24, normalized size = 0.83 \begin{align*} - \frac{\log{\left (x \right )}}{a b + b^{2} x} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09642, size = 49, normalized size = 1.69 \begin{align*} -\frac{\log \left (x\right )}{{\left (b x + a\right )} b} + \frac{\log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/((b*x + a)*b) + log(abs(-a/(b*x + a) + 1))/(a*b)