3.1.100 \(\int x (a+b \log (c x^n))^2 \log (d (e+f x^2)^m) \, dx\) [100]

Optimal. Leaf size=310 \[ -\frac {3}{4} b^2 m n^2 x^2+b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 e m n^2 \log \left (e+f x^2\right )}{4 f}+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {b e m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^2}{e}\right )}{2 f}+\frac {e m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^2}{e}\right )}{2 f}-\frac {b^2 e m n^2 \text {Li}_2\left (-\frac {f x^2}{e}\right )}{4 f}+\frac {b e m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^2}{e}\right )}{2 f}-\frac {b^2 e m n^2 \text {Li}_3\left (-\frac {f x^2}{e}\right )}{4 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2342, 2341, 2425, 272, 45, 2393, 2375, 2438, 2395, 2421, 6724} \begin {gather*} \frac {b e m n \text {PolyLog}\left (2,-\frac {f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {b^2 e m n^2 \text {PolyLog}\left (2,-\frac {f x^2}{e}\right )}{4 f}-\frac {b^2 e m n^2 \text {PolyLog}\left (3,-\frac {f x^2}{e}\right )}{4 f}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {b e m n \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f}+\frac {e m \log \left (\frac {f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {b^2 e m n^2 \log \left (e+f x^2\right )}{4 f}-\frac {3}{4} b^2 m n^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2341

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 2342

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 2375

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 2393

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 2395

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2425

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 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 6724

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.16, size = 814, normalized size = 2.63 \begin {gather*} \frac {-2 a^2 f m x^2+4 a b f m n x^2-3 b^2 f m n^2 x^2-4 a b f m x^2 \log \left (c x^n\right )+4 b^2 f m n x^2 \log \left (c x^n\right )-2 b^2 f m x^2 \log ^2\left (c x^n\right )+4 a b e m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 a b e m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-2 b^2 e m n^2 \log ^2(x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 a^2 e m \log \left (e+f x^2\right )-2 a b e m n \log \left (e+f x^2\right )+b^2 e m n^2 \log \left (e+f x^2\right )-4 a b e m n \log (x) \log \left (e+f x^2\right )+2 b^2 e m n^2 \log (x) \log \left (e+f x^2\right )+2 b^2 e m n^2 \log ^2(x) \log \left (e+f x^2\right )+4 a b e m \log \left (c x^n\right ) \log \left (e+f x^2\right )-2 b^2 e m n \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (e+f x^2\right )+2 b^2 e m \log ^2\left (c x^n\right ) \log \left (e+f x^2\right )+2 a^2 f x^2 \log \left (d \left (e+f x^2\right )^m\right )-2 a b f n x^2 \log \left (d \left (e+f x^2\right )^m\right )+b^2 f n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )+4 a b f x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 b^2 f n x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 f x^2 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b e m n \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b e m n \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b^2 e m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-4 b^2 e m n^2 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*f*m*x^2 + 4*a*b*f*m*n*x^2 - 3*b^2*f*m*n^2*x^2 - 4*a*b*f*m*x^2*Log[c*x^n] + 4*b^2*f*m*n*x^2*Log[c*x^n]
- 2*b^2*f*m*x^2*Log[c*x^n]^2 + 4*a*b*e*m*n*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]*Log[1
- (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^2*e*m*n*Log[x]*Log[c*x^
n]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 4*a*b*e*m*n*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]*L
og[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 2*b^2*e*m*n^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 4*b^2*e*m*n*Log[x]*Log
[c*x^n]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 2*a^2*e*m*Log[e + f*x^2] - 2*a*b*e*m*n*Log[e + f*x^2] + b^2*e*m*n^2*L
og[e + f*x^2] - 4*a*b*e*m*n*Log[x]*Log[e + f*x^2] + 2*b^2*e*m*n^2*Log[x]*Log[e + f*x^2] + 2*b^2*e*m*n^2*Log[x]
^2*Log[e + f*x^2] + 4*a*b*e*m*Log[c*x^n]*Log[e + f*x^2] - 2*b^2*e*m*n*Log[c*x^n]*Log[e + f*x^2] - 4*b^2*e*m*n*
Log[x]*Log[c*x^n]*Log[e + f*x^2] + 2*b^2*e*m*Log[c*x^n]^2*Log[e + f*x^2] + 2*a^2*f*x^2*Log[d*(e + f*x^2)^m] -
2*a*b*f*n*x^2*Log[d*(e + f*x^2)^m] + b^2*f*n^2*x^2*Log[d*(e + f*x^2)^m] + 4*a*b*f*x^2*Log[c*x^n]*Log[d*(e + f*
x^2)^m] - 2*b^2*f*n*x^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 2*b^2*f*x^2*Log[c*x^n]^2*Log[d*(e + f*x^2)^m] + 2*b*
e*m*n*(2*a - b*n + 2*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 2*b*e*m*n*(2*a - b*n + 2*b*Log[c*x^n
])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - 4*b^2*e*m*n^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 4*b^2*e*m*n^2*Poly
Log[3, (I*Sqrt[f]*x)/Sqrt[e]])/(4*f)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 12230, normalized size = 39.45

method	result	size
risch	\(\text {Expression too large to display}\)	\(12230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________