3.2.50 \(\int (g x)^{-1+3 m} (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\) [150]

Optimal. Leaf size=433 \[ \frac {2 b k n (g x)^{3 m}}{27 g m^2}+\frac {4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac {5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{3 f^3 g m^2} \]

[Out]

2/27*b*k*n*(g*x)^(3*m)/g/m^2+4/9*b*e^2*k*n*(g*x)^(3*m)/f^2/g/m^2/(x^(2*m))-5/36*b*e*k*n*(g*x)^(3*m)/f/g/m^2/(x
^m)-1/9*k*(g*x)^(3*m)*(a+b*ln(c*x^n))/g/m-1/3*e^2*k*(g*x)^(3*m)*(a+b*ln(c*x^n))/f^2/g/m/(x^(2*m))+1/6*e*k*(g*x
)^(3*m)*(a+b*ln(c*x^n))/f/g/m/(x^m)-1/9*b*e^3*k*n*(g*x)^(3*m)*ln(e+f*x^m)/f^3/g/m^2/(x^(3*m))-1/3*b*e^3*k*n*(g
*x)^(3*m)*ln(-f*x^m/e)*ln(e+f*x^m)/f^3/g/m^2/(x^(3*m))+1/3*e^3*k*(g*x)^(3*m)*(a+b*ln(c*x^n))*ln(e+f*x^m)/f^3/g
/m/(x^(3*m))-1/9*b*n*(g*x)^(3*m)*ln(d*(e+f*x^m)^k)/g/m^2+1/3*(g*x)^(3*m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k)/g/m
-1/3*b*e^3*k*n*(g*x)^(3*m)*polylog(2,1+f*x^m/e)/f^3/g/m^2/(x^(3*m))

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2505, 20, 272, 45, 2423, 16, 32, 30, 19, 2504, 2441, 2352} \begin {gather*} -\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \text {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{3 f^3 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac {5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}+\frac {2 b k n (g x)^{3 m}}{27 g m^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g*x)^(-1 + 3*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

(2*b*k*n*(g*x)^(3*m))/(27*g*m^2) + (4*b*e^2*k*n*(g*x)^(3*m))/(9*f^2*g*m^2*x^(2*m)) - (5*b*e*k*n*(g*x)^(3*m))/(
36*f*g*m^2*x^m) - (k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(9*g*m) - (e^2*k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(3*f^2*g
*m*x^(2*m)) + (e*k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(6*f*g*m*x^m) - (b*e^3*k*n*(g*x)^(3*m)*Log[e + f*x^m])/(9*f
^3*g*m^2*x^(3*m)) - (b*e^3*k*n*(g*x)^(3*m)*Log[-((f*x^m)/e)]*Log[e + f*x^m])/(3*f^3*g*m^2*x^(3*m)) + (e^3*k*(g
*x)^(3*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/(3*f^3*g*m*x^(3*m)) - (b*n*(g*x)^(3*m)*Log[d*(e + f*x^m)^k])/(9*g
*m^2) + ((g*x)^(3*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(3*g*m) - (b*e^3*k*n*(g*x)^(3*m)*PolyLog[2, 1 +
(f*x^m)/e])/(3*f^3*g*m^2*x^(3*m))

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]

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 20

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 32

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

Rule 45

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])

Rule 272

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 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 2423

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 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 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 2505

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]

Rubi steps

\begin {align*} \int (g x)^{-1+3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-(b n) \int \left (-\frac {k (g x)^{3 m}}{9 g m x}-\frac {e^2 k x^{-1-2 m} (g x)^{3 m}}{3 f^2 g m}+\frac {e k x^{-1-m} (g x)^{3 m}}{6 f g m}+\frac {e^3 k x^{-1-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{3 f^3 g m}+\frac {(g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{3 g m x}\right ) \, dx\\ &=-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {(b n) \int \frac {(g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{3 g m}+\frac {(b k n) \int \frac {(g x)^{3 m}}{x} \, dx}{9 g m}-\frac {\left (b e^3 k n\right ) \int x^{-1-3 m} (g x)^{3 m} \log \left (e+f x^m\right ) \, dx}{3 f^3 g m}+\frac {\left (b e^2 k n\right ) \int x^{-1-2 m} (g x)^{3 m} \, dx}{3 f^2 g m}-\frac {(b e k n) \int x^{-1-m} (g x)^{3 m} \, dx}{6 f g m}\\ &=-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {(b n) \int (g x)^{-1+3 m} \log \left (d \left (e+f x^m\right )^k\right ) \, dx}{3 m}+\frac {(b k n) \int (g x)^{-1+3 m} \, dx}{9 m}-\frac {\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \int \frac {\log \left (e+f x^m\right )}{x} \, dx}{3 f^3 g m}+\frac {\left (b e^2 k n x^{-3 m} (g x)^{3 m}\right ) \int x^{-1+m} \, dx}{3 f^2 g m}-\frac {\left (b e k n x^{-3 m} (g x)^{3 m}\right ) \int x^{-1+2 m} \, dx}{6 f g m}\\ &=\frac {b k n (g x)^{3 m}}{27 g m^2}+\frac {b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac {b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {(b f k n) \int \frac {x^{-1+m} (g x)^{3 m}}{e+f x^m} \, dx}{9 g m}-\frac {\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^m\right )}{3 f^3 g m^2}\\ &=\frac {b k n (g x)^{3 m}}{27 g m^2}+\frac {b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac {b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}+\frac {\left (b e^3 k n x^{-3 m} (g x)^{3 m}\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^m\right )}{3 f^2 g m^2}+\frac {\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \int \frac {x^{-1+4 m}}{e+f x^m} \, dx}{9 g m}\\ &=\frac {b k n (g x)^{3 m}}{27 g m^2}+\frac {b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac {b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{3 f^3 g m^2}+\frac {\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \text {Subst}\left (\int \frac {x^3}{e+f x} \, dx,x,x^m\right )}{9 g m^2}\\ &=\frac {b k n (g x)^{3 m}}{27 g m^2}+\frac {b e^2 k n x^{-2 m} (g x)^{3 m}}{3 f^2 g m^2}-\frac {b e k n x^{-m} (g x)^{3 m}}{12 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{3 f^3 g m^2}+\frac {\left (b f k n x^{-3 m} (g x)^{3 m}\right ) \text {Subst}\left (\int \left (\frac {e^2}{f^3}-\frac {e x}{f^2}+\frac {x^2}{f}-\frac {e^3}{f^3 (e+f x)}\right ) \, dx,x,x^m\right )}{9 g m^2}\\ &=\frac {2 b k n (g x)^{3 m}}{27 g m^2}+\frac {4 b e^2 k n x^{-2 m} (g x)^{3 m}}{9 f^2 g m^2}-\frac {5 b e k n x^{-m} (g x)^{3 m}}{36 f g m^2}-\frac {k (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{9 g m}-\frac {e^2 k x^{-2 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{3 f^2 g m}+\frac {e k x^{-m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right )}{6 f g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (e+f x^m\right )}{9 f^3 g m^2}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{3 f^3 g m^2}+\frac {e^3 k x^{-3 m} (g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{3 f^3 g m}-\frac {b n (g x)^{3 m} \log \left (d \left (e+f x^m\right )^k\right )}{9 g m^2}+\frac {(g x)^{3 m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{3 g m}-\frac {b e^3 k n x^{-3 m} (g x)^{3 m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{3 f^3 g m^2}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 410, normalized size = 0.95 \begin {gather*} \frac {x^{-3 m} (g x)^{3 m} \left (-36 a e^2 f k m x^m+48 b e^2 f k n x^m+18 a e f^2 k m x^{2 m}-15 b e f^2 k n x^{2 m}-12 a f^3 k m x^{3 m}+8 b f^3 k n x^{3 m}-36 b e^3 k m^2 n \log ^2(x)-36 b e^2 f k m x^m \log \left (c x^n\right )+18 b e f^2 k m x^{2 m} \log \left (c x^n\right )-12 b f^3 k m x^{3 m} \log \left (c x^n\right )+36 a e^3 k m \log \left (e-e x^m\right )-12 b e^3 k n \log \left (e-e x^m\right )+36 b e^3 k m \log \left (c x^n\right ) \log \left (e-e x^m\right )-36 b e^3 k n \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )+12 e^3 k m \log (x) \left (3 a m-b n+3 b m \log \left (c x^n\right )-3 b n \log \left (e-e x^m\right )+3 b n \log \left (e+f x^m\right )\right )+36 a f^3 m x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )-12 b f^3 n x^{3 m} \log \left (d \left (e+f x^m\right )^k\right )+36 b f^3 m x^{3 m} \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-36 b e^3 k n \text {Li}_2\left (1+\frac {f x^m}{e}\right )\right )}{108 f^3 g m^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(g*x)^(-1 + 3*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

((g*x)^(3*m)*(-36*a*e^2*f*k*m*x^m + 48*b*e^2*f*k*n*x^m + 18*a*e*f^2*k*m*x^(2*m) - 15*b*e*f^2*k*n*x^(2*m) - 12*
a*f^3*k*m*x^(3*m) + 8*b*f^3*k*n*x^(3*m) - 36*b*e^3*k*m^2*n*Log[x]^2 - 36*b*e^2*f*k*m*x^m*Log[c*x^n] + 18*b*e*f
^2*k*m*x^(2*m)*Log[c*x^n] - 12*b*f^3*k*m*x^(3*m)*Log[c*x^n] + 36*a*e^3*k*m*Log[e - e*x^m] - 12*b*e^3*k*n*Log[e
- e*x^m] + 36*b*e^3*k*m*Log[c*x^n]*Log[e - e*x^m] - 36*b*e^3*k*n*Log[-((f*x^m)/e)]*Log[e + f*x^m] + 12*e^3*k*
m*Log[x]*(3*a*m - b*n + 3*b*m*Log[c*x^n] - 3*b*n*Log[e - e*x^m] + 3*b*n*Log[e + f*x^m]) + 36*a*f^3*m*x^(3*m)*L
og[d*(e + f*x^m)^k] - 12*b*f^3*n*x^(3*m)*Log[d*(e + f*x^m)^k] + 36*b*f^3*m*x^(3*m)*Log[c*x^n]*Log[d*(e + f*x^m
)^k] - 36*b*e^3*k*n*PolyLog[2, 1 + (f*x^m)/e]))/(108*f^3*g*m^2*x^(3*m))

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{-1+3 m} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^(-1+3*m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

[Out]

int((g*x)^(-1+3*m)*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^(-1+3*m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="maxima")

[Out]

1/9*(3*b*g^(3*m)*m*x^(3*m)*log(x^n) + (3*a*g^(3*m)*m + (3*g^(3*m)*m*log(c) - g^(3*m)*n)*b)*x^(3*m))*log((f*x^m
+ e)^k)/(g*m^2) + integrate(-1/9*((3*(f*g^(3*m)*k*m - 3*f*g^(3*m)*m*log(d))*a - (f*g^(3*m)*k*n - 3*(f*g^(3*m)
*k*m - 3*f*g^(3*m)*m*log(d))*log(c))*b)*x^(4*m) - 9*(b*g^(3*m)*m*log(c)*log(d) + a*g^(3*m)*m*log(d))*e^(3*m*lo
g(x) + 1) - 3*(3*b*g^(3*m)*m*e^(3*m*log(x) + 1)*log(d) - (f*g^(3*m)*k*m - 3*f*g^(3*m)*m*log(d))*b*x^(4*m))*log
(x^n))/(f*g*m*x*x^m + g*m*x*e), x)

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Fricas [A]
time = 0.38, size = 364, normalized size = 0.84 \begin {gather*} \frac {36 \, b g^{3 \, m - 1} k m n e^{3} \log \left ({\left (f x^{m} + e\right )} e^{\left (-1\right )}\right ) \log \left (x\right ) + 36 \, b g^{3 \, m - 1} k n {\rm Li}_2\left (-{\left (f x^{m} + e\right )} e^{\left (-1\right )} + 1\right ) e^{3} - 4 \, {\left (3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - 2 \, b f^{3} k n - 3 \, {\left (3 \, b f^{3} m \log \left (c\right ) + 3 \, a f^{3} m - b f^{3} n\right )} \log \left (d\right ) + 3 \, {\left (b f^{3} k m n - 3 \, b f^{3} m n \log \left (d\right )\right )} \log \left (x\right )\right )} g^{3 \, m - 1} x^{3 \, m} + 3 \, {\left (6 \, b f^{2} k m n e \log \left (x\right ) + 6 \, b f^{2} k m e \log \left (c\right ) + {\left (6 \, a f^{2} k m - 5 \, b f^{2} k n\right )} e\right )} g^{3 \, m - 1} x^{2 \, m} - 12 \, {\left (3 \, b f k m n e^{2} \log \left (x\right ) + 3 \, b f k m e^{2} \log \left (c\right ) + {\left (3 \, a f k m - 4 \, b f k n\right )} e^{2}\right )} g^{3 \, m - 1} x^{m} + 12 \, {\left ({\left (3 \, b f^{3} k m n \log \left (x\right ) + 3 \, b f^{3} k m \log \left (c\right ) + 3 \, a f^{3} k m - b f^{3} k n\right )} g^{3 \, m - 1} x^{3 \, m} + {\left (3 \, b k m e^{3} \log \left (c\right ) + {\left (3 \, a k m - b k n\right )} e^{3}\right )} g^{3 \, m - 1}\right )} \log \left (f x^{m} + e\right )}{108 \, f^{3} m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^(-1+3*m)*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="fricas")

[Out]

1/108*(36*b*g^(3*m - 1)*k*m*n*e^3*log((f*x^m + e)*e^(-1))*log(x) + 36*b*g^(3*m - 1)*k*n*dilog(-(f*x^m + e)*e^(
-1) + 1)*e^3 - 4*(3*b*f^3*k*m*log(c) + 3*a*f^3*k*m - 2*b*f^3*k*n - 3*(3*b*f^3*m*log(c) + 3*a*f^3*m - b*f^3*n)*
log(d) + 3*(b*f^3*k*m*n - 3*b*f^3*m*n*log(d))*log(x))*g^(3*m - 1)*x^(3*m) + 3*(6*b*f^2*k*m*n*e*log(x) + 6*b*f^
2*k*m*e*log(c) + (6*a*f^2*k*m - 5*b*f^2*k*n)*e)*g^(3*m - 1)*x^(2*m) - 12*(3*b*f*k*m*n*e^2*log(x) + 3*b*f*k*m*e
^2*log(c) + (3*a*f*k*m - 4*b*f*k*n)*e^2)*g^(3*m - 1)*x^m + 12*((3*b*f^3*k*m*n*log(x) + 3*b*f^3*k*m*log(c) + 3*
a*f^3*k*m - b*f^3*k*n)*g^(3*m - 1)*x^(3*m) + (3*b*k*m*e^3*log(c) + (3*a*k*m - b*k*n)*e^3)*g^(3*m - 1))*log(f*x
^m + e))/(f^3*m^2)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)**(-1+3*m)*(a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6437 deep

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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((g*x)^(-1+3*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)^(3*m - 1)*log((f*x^m + e)^k*d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,{\left (g\,x\right )}^{3\,m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(d*(e + f*x^m)^k)*(g*x)^(3*m - 1)*(a + b*log(c*x^n)),x)

[Out]

int(log(d*(e + f*x^m)^k)*(g*x)^(3*m - 1)*(a + b*log(c*x^n)), x)

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