3.110 \(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^3} \, dx\)

Optimal. Leaf size=451 \[ -\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}+\frac {3 b^2 f m n^2 \text {Li}_2\left (-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {3 b^2 f m n^2 \text {Li}_3\left (-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b^2 f m n^2 \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {3 b f m n \text {Li}_2\left (-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {3 b f m n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}+\frac {3 b^3 f m n^3 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^3 f m n^3 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^3 f m n^3 \text {Li}_4\left (-\frac {e}{f x^2}\right )}{8 e}-\frac {3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}+\frac {3 b^3 f m n^3 \log (x)}{4 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.57, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2305, 2304, 2378, 266, 36, 29, 31, 2345, 2391, 2374, 6589, 2383} \[ \frac {3 b^2 f m n^2 \text {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {3 b^2 f m n^2 \text {PolyLog}\left (3,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac {3 b f m n \text {PolyLog}\left (2,-\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3 b^3 f m n^3 \text {PolyLog}\left (2,-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^3 f m n^3 \text {PolyLog}\left (3,-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^3 f m n^3 \text {PolyLog}\left (4,-\frac {e}{f x^2}\right )}{8 e}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b^2 f m n^2 \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac {3 b f m n \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (\frac {e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}+\frac {3 b^3 f m n^3 \log (x)}{4 e} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^3,x]

[Out]

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

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 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 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 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 2345

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> -Simp[(Log[1 +
d/(e*x^r)]*(a + b*Log[c*x^n])^p)/(d*r), x] + Dist[(b*n*p)/(d*r), Int[(Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p
- 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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 )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^3} \, dx &=-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-(2 f m) \int \left (-\frac {3 b^3 n^3}{8 x \left (e+f x^2\right )}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x \left (e+f x^2\right )}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x \left (e+f x^2\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x \left (e+f x^2\right )} \, dx+\frac {1}{2} (3 b f m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (e+f x^2\right )} \, dx+\frac {1}{2} \left (3 b^2 f m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x \left (e+f x^2\right )} \, dx+\frac {1}{4} \left (3 b^3 f m n^3\right ) \int \frac {1}{x \left (e+f x^2\right )} \, dx\\ &=-\frac {3 b^2 f m n^2 \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {(3 b f m n) \int \frac {\log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e}+\frac {\left (3 b^2 f m n^2\right ) \int \frac {\log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e}+\frac {1}{8} \left (3 b^3 f m n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x (e+f x)} \, dx,x,x^2\right )+\frac {\left (3 b^3 f m n^3\right ) \int \frac {\log \left (1+\frac {e}{f x^2}\right )}{x} \, dx}{4 e}\\ &=-\frac {3 b^2 f m n^2 \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {3 b^3 f m n^3 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}-\frac {\left (3 b^2 f m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x^2}\right )}{x} \, dx}{2 e}+\frac {\left (3 b^3 f m n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{8 e}-\frac {\left (3 b^3 f m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {e}{f x^2}\right )}{x} \, dx}{4 e}-\frac {\left (3 b^3 f^2 m n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+f x} \, dx,x,x^2\right )}{8 e}\\ &=\frac {3 b^3 f m n^3 \log (x)}{4 e}-\frac {3 b^2 f m n^2 \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {3 b^3 f m n^3 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b^3 f m n^3 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {e}{f x^2}\right )}{4 e}-\frac {\left (3 b^3 f m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {e}{f x^2}\right )}{x} \, dx}{4 e}\\ &=\frac {3 b^3 f m n^3 \log (x)}{4 e}-\frac {3 b^2 f m n^2 \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {3 b f m n \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac {f m \log \left (1+\frac {e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3 b^3 f m n^3 \log \left (e+f x^2\right )}{8 e}-\frac {3 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac {3 b^3 f m n^3 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b^3 f m n^3 \text {Li}_3\left (-\frac {e}{f x^2}\right )}{8 e}+\frac {3 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {e}{f x^2}\right )}{4 e}+\frac {3 b^3 f m n^3 \text {Li}_4\left (-\frac {e}{f x^2}\right )}{8 e}\\ \end {align*}

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Mathematica [C]  time = 0.91, size = 2248, normalized size = 4.98 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^3,x]

[Out]

-1/8*(-8*a^3*f*m*x^2*Log[x] - 12*a^2*b*f*m*n*x^2*Log[x] - 12*a*b^2*f*m*n^2*x^2*Log[x] - 6*b^3*f*m*n^3*x^2*Log[
x] + 12*a^2*b*f*m*n*x^2*Log[x]^2 + 12*a*b^2*f*m*n^2*x^2*Log[x]^2 + 6*b^3*f*m*n^3*x^2*Log[x]^2 - 8*a*b^2*f*m*n^
2*x^2*Log[x]^3 - 4*b^3*f*m*n^3*x^2*Log[x]^3 + 2*b^3*f*m*n^3*x^2*Log[x]^4 - 24*a^2*b*f*m*x^2*Log[x]*Log[c*x^n]
- 24*a*b^2*f*m*n*x^2*Log[x]*Log[c*x^n] - 12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n] + 24*a*b^2*f*m*n*x^2*Log[x]^2*Lo
g[c*x^n] + 12*b^3*f*m*n^2*x^2*Log[x]^2*Log[c*x^n] - 8*b^3*f*m*n^2*x^2*Log[x]^3*Log[c*x^n] - 24*a*b^2*f*m*x^2*L
og[x]*Log[c*x^n]^2 - 12*b^3*f*m*n*x^2*Log[x]*Log[c*x^n]^2 + 12*b^3*f*m*n*x^2*Log[x]^2*Log[c*x^n]^2 - 8*b^3*f*m
*x^2*Log[x]*Log[c*x^n]^3 + 12*a^2*b*f*m*n*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*f*m*n^2*x^2*Log
[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^3*f*m*n^3*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2*f*m*n^
2*x^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^3*f*m*n^3*x^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*
b^3*f*m*n^3*x^2*Log[x]^3*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 24*a*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt
[f]*x)/Sqrt[e]] + 12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^3*f*m*n^2*x^2*Log
[x]^2*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^3*f*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[1 - (I*Sqrt[f]*x)/S
qrt[e]] + 12*a^2*b*f*m*n*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*f*m*n^2*x^2*Log[x]*Log[1 + (I*Sq
rt[f]*x)/Sqrt[e]] + 6*b^3*f*m*n^3*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 12*a*b^2*f*m*n^2*x^2*Log[x]^2*Lo
g[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^3*f*m*n^3*x^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^3*f*m*n^3*x^2*L
og[x]^3*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 24*a*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] +
12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^3*f*m*n^2*x^2*Log[x]^2*Log[c*x^n]*
Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^3*f*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 4*a^3*f
*m*x^2*Log[e + f*x^2] + 6*a^2*b*f*m*n*x^2*Log[e + f*x^2] + 6*a*b^2*f*m*n^2*x^2*Log[e + f*x^2] + 3*b^3*f*m*n^3*
x^2*Log[e + f*x^2] - 12*a^2*b*f*m*n*x^2*Log[x]*Log[e + f*x^2] - 12*a*b^2*f*m*n^2*x^2*Log[x]*Log[e + f*x^2] - 6
*b^3*f*m*n^3*x^2*Log[x]*Log[e + f*x^2] + 12*a*b^2*f*m*n^2*x^2*Log[x]^2*Log[e + f*x^2] + 6*b^3*f*m*n^3*x^2*Log[
x]^2*Log[e + f*x^2] - 4*b^3*f*m*n^3*x^2*Log[x]^3*Log[e + f*x^2] + 12*a^2*b*f*m*x^2*Log[c*x^n]*Log[e + f*x^2] +
12*a*b^2*f*m*n*x^2*Log[c*x^n]*Log[e + f*x^2] + 6*b^3*f*m*n^2*x^2*Log[c*x^n]*Log[e + f*x^2] - 24*a*b^2*f*m*n*x
^2*Log[x]*Log[c*x^n]*Log[e + f*x^2] - 12*b^3*f*m*n^2*x^2*Log[x]*Log[c*x^n]*Log[e + f*x^2] + 12*b^3*f*m*n^2*x^2
*Log[x]^2*Log[c*x^n]*Log[e + f*x^2] + 12*a*b^2*f*m*x^2*Log[c*x^n]^2*Log[e + f*x^2] + 6*b^3*f*m*n*x^2*Log[c*x^n
]^2*Log[e + f*x^2] - 12*b^3*f*m*n*x^2*Log[x]*Log[c*x^n]^2*Log[e + f*x^2] + 4*b^3*f*m*x^2*Log[c*x^n]^3*Log[e +
f*x^2] + 4*a^3*e*Log[d*(e + f*x^2)^m] + 6*a^2*b*e*n*Log[d*(e + f*x^2)^m] + 6*a*b^2*e*n^2*Log[d*(e + f*x^2)^m]
+ 3*b^3*e*n^3*Log[d*(e + f*x^2)^m] + 12*a^2*b*e*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 12*a*b^2*e*n*Log[c*x^n]*Log[
d*(e + f*x^2)^m] + 6*b^3*e*n^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 12*a*b^2*e*Log[c*x^n]^2*Log[d*(e + f*x^2)^m]
+ 6*b^3*e*n*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + 4*b^3*e*Log[c*x^n]^3*Log[d*(e + f*x^2)^m] + 6*b*f*m*n*x^2*(2*a
^2 + 2*a*b*n + b^2*n^2 + 2*b*(2*a + b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]]
+ 6*b*f*m*n*x^2*(2*a^2 + 2*a*b*n + b^2*n^2 + 2*b*(2*a + b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, (I*S
qrt[f]*x)/Sqrt[e]] - 24*a*b^2*f*m*n^2*x^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 12*b^3*f*m*n^3*x^2*PolyLog[3,
((-I)*Sqrt[f]*x)/Sqrt[e]] - 24*b^3*f*m*n^2*x^2*Log[c*x^n]*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 24*a*b^2*f*m
*n^2*x^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^3*f*m*n^3*x^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] - 24*b^3*f*m
*n^2*x^2*Log[c*x^n]*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]] + 24*b^3*f*m*n^3*x^2*PolyLog[4, ((-I)*Sqrt[f]*x)/Sqrt[e]
] + 24*b^3*f*m*n^3*x^2*PolyLog[4, (I*Sqrt[f]*x)/Sqrt[e]])/(e*x^2)

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fricas [F]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^3,x, algorithm="fricas")

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log((f*x^2 + e)^m*d)/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + e)^m*d)/x^3, x)

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maple [F]  time = 11.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)^3*ln(d*(f*x^2+e)^m)/x^3,x)

[Out]

int((b*ln(c*x^n)+a)^3*ln(d*(f*x^2+e)^m)/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (4 \, b^{3} m \log \left (x^{n}\right )^{3} + 6 \, {\left (m n + 2 \, m \log \relax (c)\right )} a^{2} b + 6 \, {\left (m n^{2} + 2 \, m n \log \relax (c) + 2 \, m \log \relax (c)^{2}\right )} a b^{2} + {\left (3 \, m n^{3} + 6 \, m n^{2} \log \relax (c) + 6 \, m n \log \relax (c)^{2} + 4 \, m \log \relax (c)^{3}\right )} b^{3} + 4 \, a^{3} m + 6 \, {\left ({\left (m n + 2 \, m \log \relax (c)\right )} b^{3} + 2 \, a b^{2} m\right )} \log \left (x^{n}\right )^{2} + 6 \, {\left (2 \, {\left (m n + 2 \, m \log \relax (c)\right )} a b^{2} + {\left (m n^{2} + 2 \, m n \log \relax (c) + 2 \, m \log \relax (c)^{2}\right )} b^{3} + 2 \, a^{2} b m\right )} \log \left (x^{n}\right )\right )} \log \left (f x^{2} + e\right )}{8 \, x^{2}} + \int \frac {4 \, b^{3} e \log \relax (c)^{3} \log \relax (d) + 12 \, a b^{2} e \log \relax (c)^{2} \log \relax (d) + 12 \, a^{2} b e \log \relax (c) \log \relax (d) + 4 \, a^{3} e \log \relax (d) + 4 \, {\left ({\left (f m + f \log \relax (d)\right )} b^{3} x^{2} + b^{3} e \log \relax (d)\right )} \log \left (x^{n}\right )^{3} + {\left (4 \, {\left (f m + f \log \relax (d)\right )} a^{3} + 6 \, {\left (f m n + 2 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)\right )} a^{2} b + 6 \, {\left (f m n^{2} + 2 \, f m n \log \relax (c) + 2 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)^{2}\right )} a b^{2} + {\left (3 \, f m n^{3} + 6 \, f m n^{2} \log \relax (c) + 6 \, f m n \log \relax (c)^{2} + 4 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)^{3}\right )} b^{3}\right )} x^{2} + 6 \, {\left (2 \, b^{3} e \log \relax (c) \log \relax (d) + 2 \, a b^{2} e \log \relax (d) + {\left (2 \, {\left (f m + f \log \relax (d)\right )} a b^{2} + {\left (f m n + 2 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )^{2} + 6 \, {\left (2 \, b^{3} e \log \relax (c)^{2} \log \relax (d) + 4 \, a b^{2} e \log \relax (c) \log \relax (d) + 2 \, a^{2} b e \log \relax (d) + {\left (2 \, {\left (f m + f \log \relax (d)\right )} a^{2} b + 2 \, {\left (f m n + 2 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)\right )} a b^{2} + {\left (f m n^{2} + 2 \, f m n \log \relax (c) + 2 \, {\left (f m + f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )}{4 \, {\left (f x^{5} + e x^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^3,x, algorithm="maxima")

[Out]

-1/8*(4*b^3*m*log(x^n)^3 + 6*(m*n + 2*m*log(c))*a^2*b + 6*(m*n^2 + 2*m*n*log(c) + 2*m*log(c)^2)*a*b^2 + (3*m*n
^3 + 6*m*n^2*log(c) + 6*m*n*log(c)^2 + 4*m*log(c)^3)*b^3 + 4*a^3*m + 6*((m*n + 2*m*log(c))*b^3 + 2*a*b^2*m)*lo
g(x^n)^2 + 6*(2*(m*n + 2*m*log(c))*a*b^2 + (m*n^2 + 2*m*n*log(c) + 2*m*log(c)^2)*b^3 + 2*a^2*b*m)*log(x^n))*lo
g(f*x^2 + e)/x^2 + integrate(1/4*(4*b^3*e*log(c)^3*log(d) + 12*a*b^2*e*log(c)^2*log(d) + 12*a^2*b*e*log(c)*log
(d) + 4*a^3*e*log(d) + 4*((f*m + f*log(d))*b^3*x^2 + b^3*e*log(d))*log(x^n)^3 + (4*(f*m + f*log(d))*a^3 + 6*(f
*m*n + 2*(f*m + f*log(d))*log(c))*a^2*b + 6*(f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m + f*log(d))*log(c)^2)*a*b^2 + (
3*f*m*n^3 + 6*f*m*n^2*log(c) + 6*f*m*n*log(c)^2 + 4*(f*m + f*log(d))*log(c)^3)*b^3)*x^2 + 6*(2*b^3*e*log(c)*lo
g(d) + 2*a*b^2*e*log(d) + (2*(f*m + f*log(d))*a*b^2 + (f*m*n + 2*(f*m + f*log(d))*log(c))*b^3)*x^2)*log(x^n)^2
+ 6*(2*b^3*e*log(c)^2*log(d) + 4*a*b^2*e*log(c)*log(d) + 2*a^2*b*e*log(d) + (2*(f*m + f*log(d))*a^2*b + 2*(f*
m*n + 2*(f*m + f*log(d))*log(c))*a*b^2 + (f*m*n^2 + 2*f*m*n*log(c) + 2*(f*m + f*log(d))*log(c)^2)*b^3)*x^2)*lo
g(x^n))/(f*x^5 + e*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]

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))^3)/x^3,x)

[Out]

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^3)/x^3, 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))**3*ln(d*(f*x**2+e)**m)/x**3,x)

[Out]

Timed out

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