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

Optimal. Leaf size=30 \[ \frac{p}{x}-\frac{\left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0211428, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac{p}{x}-\frac{\left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p]/x^2,x]

[Out]

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0044495, size = 30, normalized size = 1. \[ \frac{p}{x}-\frac{\left (a+\frac{b}{x}\right ) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p]/x^2,x]

[Out]

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

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Maple [A]  time = 0.058, size = 48, normalized size = 1.6 \begin{align*} -{\frac{a}{b}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }-{\frac{1}{x}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }+{\frac{ap}{b}}+{\frac{p}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08494, size = 68, normalized size = 2.27 \begin{align*} -b p{\left (\frac{a \log \left (a x + b\right )}{b^{2}} - \frac{a \log \left (x\right )}{b^{2}} - \frac{1}{b x}\right )} - \frac{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*p*(a*log(a*x + b)/b^2 - a*log(x)/b^2 - 1/(b*x)) - log((a + b/x)^p*c)/x

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Fricas [A]  time = 2.15596, size = 77, normalized size = 2.57 \begin{align*} \frac{b p - b \log \left (c\right ) -{\left (a p x + b p\right )} \log \left (\frac{a x + b}{x}\right )}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.92736, size = 39, normalized size = 1.3 \begin{align*} \begin{cases} - \frac{a p \log{\left (a + \frac{b}{x} \right )}}{b} - \frac{p \log{\left (a + \frac{b}{x} \right )}}{x} + \frac{p}{x} - \frac{\log{\left (c \right )}}{x} & \text{for}\: b \neq 0 \\- \frac{\log{\left (a^{p} c \right )}}{x} & \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**2,x)

[Out]

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

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Giac [A]  time = 1.33727, size = 69, normalized size = 2.3 \begin{align*} -\frac{a p \log \left (a x + b\right )}{b} + \frac{a p \log \left (x\right )}{b} - \frac{p \log \left (a x + b\right )}{x} + \frac{p \log \left (x\right )}{x} + \frac{p - \log \left (c\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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