3.153 \(\int (g x)^{-1-m} (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\)

Optimal. Leaf size=304 \[ \frac{b f k n x^m (g x)^{-m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{e g m^2}-\frac{(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac{f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac{f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-\frac{b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac{b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac{b f k n x^m (g x)^{-m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac{b f k n x^m \log ^2(x) (g x)^{-m}}{2 e g}+\frac{b f k n x^m \log (x) (g x)^{-m}}{e g m} \]

[Out]

(b*f*k*n*x^m*Log[x])/(e*g*m*(g*x)^m) - (b*f*k*n*x^m*Log[x]^2)/(2*e*g*(g*x)^m) + (f*k*x^m*Log[x]*(a + b*Log[c*x
^n]))/(e*g*(g*x)^m) - (b*f*k*n*x^m*Log[e + f*x^m])/(e*g*m^2*(g*x)^m) + (b*f*k*n*x^m*Log[-((f*x^m)/e)]*Log[e +
f*x^m])/(e*g*m^2*(g*x)^m) - (f*k*x^m*(a + b*Log[c*x^n])*Log[e + f*x^m])/(e*g*m*(g*x)^m) - (b*n*Log[d*(e + f*x^
m)^k])/(g*m^2*(g*x)^m) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(g*m*(g*x)^m) + (b*f*k*n*x^m*PolyLog[2, 1 +
(f*x^m)/e])/(e*g*m^2*(g*x)^m)

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Rubi [A]  time = 0.305099, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2455, 19, 266, 36, 29, 31, 2376, 2301, 2454, 2394, 2315, 16} \[ \frac{b f k n x^m (g x)^{-m} \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right )}{e g m^2}-\frac{(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac{f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac{f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-\frac{b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac{b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac{b f k n x^m (g x)^{-m} \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac{b f k n x^m \log ^2(x) (g x)^{-m}}{2 e g}+\frac{b f k n x^m \log (x) (g x)^{-m}}{e g m} \]

Antiderivative was successfully verified.

[In]

Int[(g*x)^(-1 - m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

(b*f*k*n*x^m*Log[x])/(e*g*m*(g*x)^m) - (b*f*k*n*x^m*Log[x]^2)/(2*e*g*(g*x)^m) + (f*k*x^m*Log[x]*(a + b*Log[c*x
^n]))/(e*g*(g*x)^m) - (b*f*k*n*x^m*Log[e + f*x^m])/(e*g*m^2*(g*x)^m) + (b*f*k*n*x^m*Log[-((f*x^m)/e)]*Log[e +
f*x^m])/(e*g*m^2*(g*x)^m) - (f*k*x^m*(a + b*Log[c*x^n])*Log[e + f*x^m])/(e*g*m*(g*x)^m) - (b*n*Log[d*(e + f*x^
m)^k])/(g*m^2*(g*x)^m) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(g*m*(g*x)^m) + (b*f*k*n*x^m*PolyLog[2, 1 +
(f*x^m)/e])/(e*g*m^2*(g*x)^m)

Rule 2455

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

Rule 19

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + n)*(b*v)^n)/(a*v)^n, Int[u*v^(m + n),
x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 2394

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 2315

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

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rubi steps

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

Mathematica [A]  time = 0.340227, size = 162, normalized size = 0.53 \[ \frac{(g x)^{-m} \left (-2 b f k n x^m \text{PolyLog}\left (2,-\frac{f x^m}{e}\right )-2 \left (a m+b m \log \left (c x^n\right )+b n\right ) \left (e \log \left (d \left (e+f x^m\right )^k\right )+f k x^m \log \left (f-f x^{-m}\right )\right )+2 f k m x^m \log (x) \left (a m+b m \log \left (c x^n\right )-b n \log \left (\frac{f x^m}{e}+1\right )+b n \log \left (f-f x^{-m}\right )+b n\right )-b f k m^2 n x^m \log ^2(x)\right )}{2 e g m^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(g*x)^(-1 - m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

(-(b*f*k*m^2*n*x^m*Log[x]^2) - 2*(a*m + b*n + b*m*Log[c*x^n])*(f*k*x^m*Log[f - f/x^m] + e*Log[d*(e + f*x^m)^k]
) + 2*f*k*m*x^m*Log[x]*(a*m + b*n + b*m*Log[c*x^n] + b*n*Log[f - f/x^m] - b*n*Log[1 + (f*x^m)/e]) - 2*b*f*k*n*
x^m*PolyLog[2, -((f*x^m)/e)])/(2*e*g*m^2*(g*x)^m)

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Maple [F]  time = 0.23, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{-1-m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^(-1-m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

[Out]

int((g*x)^(-1-m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.954988, size = 595, normalized size = 1.96 \begin{align*} -\frac{2 \, b f g^{-m - 1} k m n x^{m} \log \left (x\right ) \log \left (\frac{f x^{m} + e}{e}\right ) + 2 \, b f g^{-m - 1} k n x^{m}{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) -{\left (b f k m^{2} n \log \left (x\right )^{2} + 2 \,{\left (b f k m^{2} \log \left (c\right ) + a f k m^{2} + b f k m n\right )} \log \left (x\right )\right )} g^{-m - 1} x^{m} + 2 \,{\left (b e m n \log \left (d\right ) \log \left (x\right ) +{\left (b e m \log \left (c\right ) + a e m + b e n\right )} \log \left (d\right )\right )} g^{-m - 1} + 2 \,{\left ({\left (b f k m \log \left (c\right ) + a f k m + b f k n\right )} g^{-m - 1} x^{m} +{\left (b e k m n \log \left (x\right ) + b e k m \log \left (c\right ) + a e k m + b e k n\right )} g^{-m - 1}\right )} \log \left (f x^{m} + e\right )}{2 \, e m^{2} x^{m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="fricas")

[Out]

-1/2*(2*b*f*g^(-m - 1)*k*m*n*x^m*log(x)*log((f*x^m + e)/e) + 2*b*f*g^(-m - 1)*k*n*x^m*dilog(-(f*x^m + e)/e + 1
) - (b*f*k*m^2*n*log(x)^2 + 2*(b*f*k*m^2*log(c) + a*f*k*m^2 + b*f*k*m*n)*log(x))*g^(-m - 1)*x^m + 2*(b*e*m*n*l
og(d)*log(x) + (b*e*m*log(c) + a*e*m + b*e*n)*log(d))*g^(-m - 1) + 2*((b*f*k*m*log(c) + a*f*k*m + b*f*k*n)*g^(
-m - 1)*x^m + (b*e*k*m*n*log(x) + b*e*k*m*log(c) + a*e*k*m + b*e*k*n)*g^(-m - 1))*log(f*x^m + e))/(e*m^2*x^m)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)**(-1-m)*(a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{-m - 1} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^(-1-m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*(g*x)^(-m - 1)*log((f*x^m + e)^k*d), x)