3.103 \(\int \frac {(a+b \log (c x^n))^2 \log (d (e+f x^2)^m)}{x^5} \, dx\)

Optimal. Leaf size=356 \[ -\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {b f^2 m n \text {Li}_2\left (-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {f^2 m \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b f^2 m n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{16 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {7 b^2 f m n^2}{32 e x^2} \]

[Out]

-7/32*b^2*f*m*n^2/e/x^2-1/16*b^2*f^2*m*n^2*ln(x)/e^2-3/8*b*f*m*n*(a+b*ln(c*x^n))/e/x^2+1/8*b*f^2*m*n*ln(1+e/f/
x^2)*(a+b*ln(c*x^n))/e^2-1/4*f*m*(a+b*ln(c*x^n))^2/e/x^2+1/4*f^2*m*ln(1+e/f/x^2)*(a+b*ln(c*x^n))^2/e^2+1/32*b^
2*f^2*m*n^2*ln(f*x^2+e)/e^2-1/32*b^2*n^2*ln(d*(f*x^2+e)^m)/x^4-1/8*b*n*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^4-1
/4*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^4-1/16*b^2*f^2*m*n^2*polylog(2,-e/f/x^2)/e^2-1/4*b*f^2*m*n*(a+b*ln(c*
x^n))*polylog(2,-e/f/x^2)/e^2-1/8*b^2*f^2*m*n^2*polylog(3,-e/f/x^2)/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 408, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2305, 2304, 2378, 266, 44, 2351, 2301, 2337, 2391, 2353, 2302, 30, 2374, 6589} \[ \frac {b f^2 m n \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac {b^2 f^2 m n^2 \text {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{16 e^2}-\frac {b^2 f^2 m n^2 \text {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{8 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {f^2 m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac {b f^2 m n \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {7 b^2 f m n^2}{32 e x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x^5,x]

[Out]

(-7*b^2*f*m*n^2)/(32*e*x^2) - (b^2*f^2*m*n^2*Log[x])/(16*e^2) - (3*b*f*m*n*(a + b*Log[c*x^n]))/(8*e*x^2) - (f^
2*m*(a + b*Log[c*x^n])^2)/(8*e^2) - (f*m*(a + b*Log[c*x^n])^2)/(4*e*x^2) - (f^2*m*(a + b*Log[c*x^n])^3)/(6*b*e
^2*n) + (b^2*f^2*m*n^2*Log[e + f*x^2])/(32*e^2) - (b^2*n^2*Log[d*(e + f*x^2)^m])/(32*x^4) - (b*n*(a + b*Log[c*
x^n])*Log[d*(e + f*x^2)^m])/(8*x^4) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(4*x^4) + (b*f^2*m*n*(a + b*
Log[c*x^n])*Log[1 + (f*x^2)/e])/(8*e^2) + (f^2*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(4*e^2) + (b^2*f^2*m
*n^2*PolyLog[2, -((f*x^2)/e)])/(16*e^2) + (b*f^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/(4*e^2) - (b
^2*f^2*m*n^2*PolyLog[3, -((f*x^2)/e)])/(8*e^2)

Rule 30

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 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 2302

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

Rule 2391

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 6589

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{x^5} \, dx &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-(2 f m) \int \left (-\frac {b^2 n^2}{32 x^3 \left (e+f x^2\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{8 x^3 \left (e+f x^2\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{4 x^3 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{2} (f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 \left (e+f x^2\right )} \, dx+\frac {1}{4} (b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (e+f x^2\right )} \, dx+\frac {1}{16} \left (b^2 f m n^2\right ) \int \frac {1}{x^3 \left (e+f x^2\right )} \, dx\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {1}{2} (f m) \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}+\frac {f^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{4} (b f m n) \int \left (\frac {a+b \log \left (c x^n\right )}{e x^3}-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {f^2 x \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{32} \left (b^2 f m n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (e+f x)} \, dx,x,x^2\right )\\ &=-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{2 e}-\frac {\left (f^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}+\frac {\left (f^3 m\right ) \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{2 e^2}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{4 e}-\frac {\left (b f^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{4 e^2}+\frac {\left (b f^3 m n\right ) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx}{4 e^2}+\frac {1}{32} \left (b^2 f m n^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}-\frac {\left (f^2 m\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac {(b f m n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{2 e}-\frac {\left (b f^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{2 e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\log \left (1+\frac {f x^2}{e}\right )}{x} \, dx}{8 e^2}\\ &=-\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{16 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 e^2}-\frac {\left (b^2 f^2 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x^2}{e}\right )}{x} \, dx}{4 e^2}\\ &=-\frac {7 b^2 f m n^2}{32 e x^2}-\frac {b^2 f^2 m n^2 \log (x)}{16 e^2}-\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )}{8 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )^2}{4 e x^2}-\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {b^2 f^2 m n^2 \log \left (e+f x^2\right )}{32 e^2}-\frac {b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{32 x^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{8 x^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{8 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{4 e^2}+\frac {b^2 f^2 m n^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{16 e^2}+\frac {b f^2 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 e^2}-\frac {b^2 f^2 m n^2 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{8 e^2}\\ \end {align*}

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Mathematica [C]  time = 0.48, size = 1111, normalized size = 3.12 \[ -\frac {16 b^2 f^2 m n^2 \log ^3(x) x^4-12 b^2 f^2 m n^2 \log ^2(x) x^4-48 a b f^2 m n \log ^2(x) x^4+48 b^2 f^2 m \log (x) \log ^2\left (c x^n\right ) x^4+6 b^2 f^2 m n^2 \log (x) x^4+48 a^2 f^2 m \log (x) x^4+24 a b f^2 m n \log (x) x^4-48 b^2 f^2 m n \log ^2(x) \log \left (c x^n\right ) x^4+96 a b f^2 m \log (x) \log \left (c x^n\right ) x^4+24 b^2 f^2 m n \log (x) \log \left (c x^n\right ) x^4+24 b^2 f^2 m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4-12 b^2 f^2 m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4-48 a b f^2 m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4-48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+24 b^2 f^2 m n^2 \log ^2(x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) x^4-12 b^2 f^2 m n^2 \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) x^4-48 a b f^2 m n \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) x^4-48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right ) x^4-3 b^2 f^2 m n^2 \log \left (f x^2+e\right ) x^4-24 b^2 f^2 m n^2 \log ^2(x) \log \left (f x^2+e\right ) x^4-24 b^2 f^2 m \log ^2\left (c x^n\right ) \log \left (f x^2+e\right ) x^4-24 a^2 f^2 m \log \left (f x^2+e\right ) x^4-12 a b f^2 m n \log \left (f x^2+e\right ) x^4+12 b^2 f^2 m n^2 \log (x) \log \left (f x^2+e\right ) x^4+48 a b f^2 m n \log (x) \log \left (f x^2+e\right ) x^4-48 a b f^2 m \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4-12 b^2 f^2 m n \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4+48 b^2 f^2 m n \log (x) \log \left (c x^n\right ) \log \left (f x^2+e\right ) x^4-12 b f^2 m n \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4-12 b f^2 m n \left (4 a+b n+4 b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+48 b^2 f^2 m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+48 b^2 f^2 m n^2 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) x^4+21 b^2 e f m n^2 x^2+24 b^2 e f m \log ^2\left (c x^n\right ) x^2+24 a^2 e f m x^2+36 a b e f m n x^2+48 a b e f m \log \left (c x^n\right ) x^2+36 b^2 e f m n \log \left (c x^n\right ) x^2+24 a^2 e^2 \log \left (d \left (f x^2+e\right )^m\right )+3 b^2 e^2 n^2 \log \left (d \left (f x^2+e\right )^m\right )+24 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+12 a b e^2 n \log \left (d \left (f x^2+e\right )^m\right )+48 a b e^2 \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+12 b^2 e^2 n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )}{96 e^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x^5,x]

[Out]

-1/96*(24*a^2*e*f*m*x^2 + 36*a*b*e*f*m*n*x^2 + 21*b^2*e*f*m*n^2*x^2 + 48*a^2*f^2*m*x^4*Log[x] + 24*a*b*f^2*m*n
*x^4*Log[x] + 6*b^2*f^2*m*n^2*x^4*Log[x] - 48*a*b*f^2*m*n*x^4*Log[x]^2 - 12*b^2*f^2*m*n^2*x^4*Log[x]^2 + 16*b^
2*f^2*m*n^2*x^4*Log[x]^3 + 48*a*b*e*f*m*x^2*Log[c*x^n] + 36*b^2*e*f*m*n*x^2*Log[c*x^n] + 96*a*b*f^2*m*x^4*Log[
x]*Log[c*x^n] + 24*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n] - 48*b^2*f^2*m*n*x^4*Log[x]^2*Log[c*x^n] + 24*b^2*e*f*m*x
^2*Log[c*x^n]^2 + 48*b^2*f^2*m*x^4*Log[x]*Log[c*x^n]^2 - 48*a*b*f^2*m*n*x^4*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[
e]] - 12*b^2*f^2*m*n^2*x^4*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 24*b^2*f^2*m*n^2*x^4*Log[x]^2*Log[1 - (I*Sq
rt[f]*x)/Sqrt[e]] - 48*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 48*a*b*f^2*m*n*x^4*L
og[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^2*f^2*m*n^2*x^4*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 24*b^2*f^2
*m*n^2*x^4*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 48*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x
)/Sqrt[e]] - 24*a^2*f^2*m*x^4*Log[e + f*x^2] - 12*a*b*f^2*m*n*x^4*Log[e + f*x^2] - 3*b^2*f^2*m*n^2*x^4*Log[e +
f*x^2] + 48*a*b*f^2*m*n*x^4*Log[x]*Log[e + f*x^2] + 12*b^2*f^2*m*n^2*x^4*Log[x]*Log[e + f*x^2] - 24*b^2*f^2*m
*n^2*x^4*Log[x]^2*Log[e + f*x^2] - 48*a*b*f^2*m*x^4*Log[c*x^n]*Log[e + f*x^2] - 12*b^2*f^2*m*n*x^4*Log[c*x^n]*
Log[e + f*x^2] + 48*b^2*f^2*m*n*x^4*Log[x]*Log[c*x^n]*Log[e + f*x^2] - 24*b^2*f^2*m*x^4*Log[c*x^n]^2*Log[e + f
*x^2] + 24*a^2*e^2*Log[d*(e + f*x^2)^m] + 12*a*b*e^2*n*Log[d*(e + f*x^2)^m] + 3*b^2*e^2*n^2*Log[d*(e + f*x^2)^
m] + 48*a*b*e^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 12*b^2*e^2*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 24*b^2*e^2*Lo
g[c*x^n]^2*Log[d*(e + f*x^2)^m] - 12*b*f^2*m*n*x^4*(4*a + b*n + 4*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sq
rt[e]] - 12*b*f^2*m*n*x^4*(4*a + b*n + 4*b*Log[c*x^n])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] + 48*b^2*f^2*m*n^2*x^
4*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 48*b^2*f^2*m*n^2*x^4*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(e^2*x^4)

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fricas [F]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^5,x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*log((f*x^2 + e)^m*d)/x^5, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^5,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*x^2 + e)^m*d)/x^5, x)

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maple [F]  time = 5.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)^2*ln(d*(f*x^2+e)^m)/x^5,x)

[Out]

int((b*ln(c*x^n)+a)^2*ln(d*(f*x^2+e)^m)/x^5,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (8 \, b^{2} m \log \left (x^{n}\right )^{2} + 4 \, {\left (m n + 4 \, m \log \relax (c)\right )} a b + {\left (m n^{2} + 4 \, m n \log \relax (c) + 8 \, m \log \relax (c)^{2}\right )} b^{2} + 8 \, a^{2} m + 4 \, {\left ({\left (m n + 4 \, m \log \relax (c)\right )} b^{2} + 4 \, a b m\right )} \log \left (x^{n}\right )\right )} \log \left (f x^{2} + e\right )}{32 \, x^{4}} + \int \frac {16 \, b^{2} e \log \relax (c)^{2} \log \relax (d) + 32 \, a b e \log \relax (c) \log \relax (d) + 16 \, a^{2} e \log \relax (d) + {\left (8 \, {\left (f m + 2 \, f \log \relax (d)\right )} a^{2} + 4 \, {\left (f m n + 4 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (f m n^{2} + 4 \, f m n \log \relax (c) + 8 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{2} + 8 \, {\left ({\left (f m + 2 \, f \log \relax (d)\right )} b^{2} x^{2} + 2 \, b^{2} e \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 4 \, {\left (8 \, b^{2} e \log \relax (c) \log \relax (d) + 8 \, a b e \log \relax (d) + {\left (4 \, {\left (f m + 2 \, f \log \relax (d)\right )} a b + {\left (f m n + 4 \, {\left (f m + 2 \, f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (x^{n}\right )}{16 \, {\left (f x^{7} + e x^{5}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*log(d*(f*x^2+e)^m)/x^5,x, algorithm="maxima")

[Out]

-1/32*(8*b^2*m*log(x^n)^2 + 4*(m*n + 4*m*log(c))*a*b + (m*n^2 + 4*m*n*log(c) + 8*m*log(c)^2)*b^2 + 8*a^2*m + 4
*((m*n + 4*m*log(c))*b^2 + 4*a*b*m)*log(x^n))*log(f*x^2 + e)/x^4 + integrate(1/16*(16*b^2*e*log(c)^2*log(d) +
32*a*b*e*log(c)*log(d) + 16*a^2*e*log(d) + (8*(f*m + 2*f*log(d))*a^2 + 4*(f*m*n + 4*(f*m + 2*f*log(d))*log(c))
*a*b + (f*m*n^2 + 4*f*m*n*log(c) + 8*(f*m + 2*f*log(d))*log(c)^2)*b^2)*x^2 + 8*((f*m + 2*f*log(d))*b^2*x^2 + 2
*b^2*e*log(d))*log(x^n)^2 + 4*(8*b^2*e*log(c)*log(d) + 8*a*b*e*log(d) + (4*(f*m + 2*f*log(d))*a*b + (f*m*n + 4
*(f*m + 2*f*log(d))*log(c))*b^2)*x^2)*log(x^n))/(f*x^7 + e*x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^2)/x^5,x)

[Out]

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^2)/x^5, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2*ln(d*(f*x**2+e)**m)/x**5,x)

[Out]

Timed out

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