3.3.44 \(\int \frac {\log (c (a+\frac {b}{x})^p)}{x (d+e x)} \, dx\) [244]

Optimal. Leaf size=159 \[ -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d} \]

[Out]

-ln(c*(a+b/x)^p)*ln(-b/a/x)/d-ln(c*(a+b/x)^p)*ln(e*x+d)/d-p*ln(-e*x/d)*ln(e*x+d)/d+p*ln(-e*(a*x+b)/(a*d-b*e))*
ln(e*x+d)/d-p*polylog(2,1+b/a/x)/d+p*polylog(2,a*(e*x+d)/(a*d-b*e))/d-p*polylog(2,1+e*x/d)/d

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Rubi [A]
time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2516, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \begin {gather*} \frac {p \text {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {PolyLog}\left (2,\frac {b}{a x}+1\right )}{d}-\frac {p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p]/(x*(d + e*x)),x]

[Out]

-((Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d) - (Log[c*(a + b/x)^p]*Log[d + e*x])/d - (p*Log[-((e*x)/d)]*Log[d + e
*x])/d + (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d - (p*PolyLog[2, 1 + b/(a*x)])/d + (p*PolyLog[2,
(a*(d + e*x))/(a*d - b*e)])/d - (p*PolyLog[2, 1 + (e*x)/d])/d

Rule 266

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

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

Rule 2441

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2504

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 2512

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

Rule 2516

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_.) + (g_.)*(x_))^(r_.), x_S
ymbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e,
f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x}\right )}{d}-\frac {(b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {(b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{d}+\frac {(b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(a p) \int \frac {\log (d+e x)}{b+a x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {(e p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 139, normalized size = 0.87 \begin {gather*} -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )+\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)+p \log \left (-\frac {e x}{d}\right ) \log (d+e x)-p \log \left (\frac {e (b+a x)}{-a d+b e}\right ) \log (d+e x)+p \text {Li}_2\left (1+\frac {b}{a x}\right )-p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )+p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p]/(x*(d + e*x)),x]

[Out]

-((Log[c*(a + b/x)^p]*Log[-(b/(a*x))] + Log[c*(a + b/x)^p]*Log[d + e*x] + p*Log[-((e*x)/d)]*Log[d + e*x] - p*L
og[(e*(b + a*x))/(-(a*d) + b*e)]*Log[d + e*x] + p*PolyLog[2, 1 + b/(a*x)] - p*PolyLog[2, (a*(d + e*x))/(a*d -
b*e)] + p*PolyLog[2, 1 + (e*x)/d])/d)

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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x \left (e x +d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 188, normalized size = 1.18 \begin {gather*} -\frac {1}{2} \, b p {\left (\frac {2 \, \log \left (x e + d\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{b d} + \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (x\right ) \log \left (\frac {x e}{d} + 1\right ) + {\rm Li}_2\left (-\frac {x e}{d}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (x e + d\right ) \log \left (-\frac {a x e + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a x e + a d}{a d - b e}\right )\right )}}{b d}\right )} - {\left (\frac {\log \left (x e + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(c*((a*x + b)/x)^p)/(x^2*e + d*x), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*(a + b/x)**p)/(x*(d + e*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((a + b/x)^p*c)/((x*e + d)*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x)^p)/(x*(d + e*x)),x)

[Out]

int(log(c*(a + b/x)^p)/(x*(d + e*x)), x)

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