3.35 \(\int \frac{\log (c (a+\frac{b}{x})^p)}{x^5} \, dx\)

Optimal. Leaf size=87 \[ \frac{a^2 p}{8 b^2 x^2}-\frac{a^3 p}{4 b^3 x}+\frac{a^4 p \log \left (a+\frac{b}{x}\right )}{4 b^4}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}-\frac{a p}{12 b x^3}+\frac{p}{16 x^4} \]

[Out]

p/(16*x^4) - (a*p)/(12*b*x^3) + (a^2*p)/(8*b^2*x^2) - (a^3*p)/(4*b^3*x) + (a^4*p*Log[a + b/x])/(4*b^4) - Log[c
*(a + b/x)^p]/(4*x^4)

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Rubi [A]  time = 0.0560602, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2395, 43} \[ \frac{a^2 p}{8 b^2 x^2}-\frac{a^3 p}{4 b^3 x}+\frac{a^4 p \log \left (a+\frac{b}{x}\right )}{4 b^4}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}-\frac{a p}{12 b x^3}+\frac{p}{16 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p]/x^5,x]

[Out]

p/(16*x^4) - (a*p)/(12*b*x^3) + (a^2*p)/(8*b^2*x^2) - (a^3*p)/(4*b^3*x) + (a^4*p*Log[a + b/x])/(4*b^4) - Log[c
*(a + b/x)^p]/(4*x^4)

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 2395

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

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{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{x^5} \, dx &=-\operatorname{Subst}\left (\int x^3 \log \left (c (a+b x)^p\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}+\frac{1}{4} (b p) \operatorname{Subst}\left (\int \frac{x^4}{a+b x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}+\frac{1}{4} (b p) \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{p}{16 x^4}-\frac{a p}{12 b x^3}+\frac{a^2 p}{8 b^2 x^2}-\frac{a^3 p}{4 b^3 x}+\frac{a^4 p \log \left (a+\frac{b}{x}\right )}{4 b^4}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0207037, size = 87, normalized size = 1. \[ \frac{a^2 p}{8 b^2 x^2}-\frac{a^3 p}{4 b^3 x}+\frac{a^4 p \log \left (a+\frac{b}{x}\right )}{4 b^4}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{4 x^4}-\frac{a p}{12 b x^3}+\frac{p}{16 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p]/x^5,x]

[Out]

p/(16*x^4) - (a*p)/(12*b*x^3) + (a^2*p)/(8*b^2*x^2) - (a^3*p)/(4*b^3*x) + (a^4*p*Log[a + b/x])/(4*b^4) - Log[c
*(a + b/x)^p]/(4*x^4)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b/x)^p)/x^5,x)

[Out]

int(ln(c*(a+b/x)^p)/x^5,x)

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Maxima [A]  time = 1.09044, size = 115, normalized size = 1.32 \begin{align*} \frac{1}{48} \, b p{\left (\frac{12 \, a^{4} \log \left (a x + b\right )}{b^{5}} - \frac{12 \, a^{4} \log \left (x\right )}{b^{5}} - \frac{12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{b^{4} x^{4}}\right )} - \frac{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/x^5,x, algorithm="maxima")

[Out]

1/48*b*p*(12*a^4*log(a*x + b)/b^5 - 12*a^4*log(x)/b^5 - (12*a^3*x^3 - 6*a^2*b*x^2 + 4*a*b^2*x - 3*b^3)/(b^4*x^
4)) - 1/4*log((a + b/x)^p*c)/x^4

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Fricas [A]  time = 2.30261, size = 181, normalized size = 2.08 \begin{align*} -\frac{12 \, a^{3} b p x^{3} - 6 \, a^{2} b^{2} p x^{2} + 4 \, a b^{3} p x - 3 \, b^{4} p + 12 \, b^{4} \log \left (c\right ) - 12 \,{\left (a^{4} p x^{4} - b^{4} p\right )} \log \left (\frac{a x + b}{x}\right )}{48 \, b^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/x^5,x, algorithm="fricas")

[Out]

-1/48*(12*a^3*b*p*x^3 - 6*a^2*b^2*p*x^2 + 4*a*b^3*p*x - 3*b^4*p + 12*b^4*log(c) - 12*(a^4*p*x^4 - b^4*p)*log((
a*x + b)/x))/(b^4*x^4)

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Sympy [A]  time = 73.2739, size = 94, normalized size = 1.08 \begin{align*} \begin{cases} \frac{a^{4} p \log{\left (a + \frac{b}{x} \right )}}{4 b^{4}} - \frac{a^{3} p}{4 b^{3} x} + \frac{a^{2} p}{8 b^{2} x^{2}} - \frac{a p}{12 b x^{3}} - \frac{p \log{\left (a + \frac{b}{x} \right )}}{4 x^{4}} + \frac{p}{16 x^{4}} - \frac{\log{\left (c \right )}}{4 x^{4}} & \text{for}\: b \neq 0 \\- \frac{\log{\left (a^{p} c \right )}}{4 x^{4}} & \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**5,x)

[Out]

Piecewise((a**4*p*log(a + b/x)/(4*b**4) - a**3*p/(4*b**3*x) + a**2*p/(8*b**2*x**2) - a*p/(12*b*x**3) - p*log(a
+ b/x)/(4*x**4) + p/(16*x**4) - log(c)/(4*x**4), Ne(b, 0)), (-log(a**p*c)/(4*x**4), True))

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Giac [A]  time = 1.19987, size = 130, normalized size = 1.49 \begin{align*} \frac{a^{4} p \log \left (a x + b\right )}{4 \, b^{4}} - \frac{a^{4} p \log \left (x\right )}{4 \, b^{4}} - \frac{p \log \left (a x + b\right )}{4 \, x^{4}} + \frac{p \log \left (x\right )}{4 \, x^{4}} - \frac{12 \, a^{3} p x^{3} - 6 \, a^{2} b p x^{2} + 4 \, a b^{2} p x - 3 \, b^{3} p + 12 \, b^{3} \log \left (c\right )}{48 \, b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/x^5,x, algorithm="giac")

[Out]

1/4*a^4*p*log(a*x + b)/b^4 - 1/4*a^4*p*log(x)/b^4 - 1/4*p*log(a*x + b)/x^4 + 1/4*p*log(x)/x^4 - 1/48*(12*a^3*p
*x^3 - 6*a^2*b*p*x^2 + 4*a*b^2*p*x - 3*b^3*p + 12*b^3*log(c))/(b^3*x^4)