3.30 \(\int \log (c (a+\frac{b}{x})^p) \, dx\)

Optimal. Leaf size=27 \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log (a x+b)}{a} \]

[Out]

x*Log[c*(a + b/x)^p] + (b*p*Log[b + a*x])/a

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Rubi [A]  time = 0.0087283, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2448, 263, 31} \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p],x]

[Out]

x*Log[c*(a + b/x)^p] + (b*p*Log[b + a*x])/a

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A]  time = 0.002082, size = 37, normalized size = 1.37 \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log \left (a+\frac{b}{x}\right )}{a}+\frac{b p \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p],x]

[Out]

(b*p*Log[a + b/x])/a + x*Log[c*(a + b/x)^p] + (b*p*Log[x])/a

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Maple [A]  time = 0.06, size = 30, normalized size = 1.1 \begin{align*} x\ln \left ( c \left ({\frac{ax+b}{x}} \right ) ^{p} \right ) +{\frac{bp\ln \left ( ax+b \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(c*((a*x+b)/x)^p)+b*p*ln(a*x+b)/a

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Maxima [A]  time = 1.14417, size = 36, normalized size = 1.33 \begin{align*} x \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) + \frac{b p \log \left (a x + b\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log((a + b/x)^p*c) + b*p*log(a*x + b)/a

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Fricas [A]  time = 2.20548, size = 81, normalized size = 3. \begin{align*} \frac{a p x \log \left (\frac{a x + b}{x}\right ) + b p \log \left (a x + b\right ) + a x \log \left (c\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*p*x*log((a*x + b)/x) + b*p*log(a*x + b) + a*x*log(c))/a

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Sympy [A]  time = 1.31805, size = 48, normalized size = 1.78 \begin{align*} \begin{cases} p x \log{\left (a + \frac{b}{x} \right )} + x \log{\left (c \right )} + \frac{b p \log{\left (a x + b \right )}}{a} & \text{for}\: a \neq 0 \\p x \log{\left (b \right )} - p x \log{\left (x \right )} + p x + x \log{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b/x)**p),x)

[Out]

Piecewise((p*x*log(a + b/x) + x*log(c) + b*p*log(a*x + b)/a, Ne(a, 0)), (p*x*log(b) - p*x*log(x) + p*x + x*log
(c), True))

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Giac [A]  time = 1.27674, size = 43, normalized size = 1.59 \begin{align*} p x \log \left (a x + b\right ) - p x \log \left (x\right ) + \frac{b p \log \left (a x + b\right )}{a} + x \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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