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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.536124, antiderivative size = 310, normalized size of antiderivative = 1., 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 = {2305, 2304, 2378, 266, 43, 2351, 2337, 2391, 2353, 2374, 6589} \[ \frac{b e m n \text{PolyLog}\left (2,-\frac{f x^2}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{b^2 e m n^2 \text{PolyLog}\left (2,-\frac{f x^2}{e}\right )}{4 f}-\frac{b^2 e m n^2 \text{PolyLog}\left (3,-\frac{f x^2}{e}\right )}{4 f}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac{b e m n \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f}+\frac{e m \log \left (\frac{f x^2}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b^2 n^2 x^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac{b^2 e m n^2 \log \left (e+f x^2\right )}{4 f}-\frac{3}{4} b^2 m n^2 x^2 \]

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 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 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 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 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 43

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 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 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 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 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 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 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 ) \operatorname{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 ) \operatorname{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]  time = 0.251569, size = 814, normalized size = 2.63 \[ \frac{-2 f m x^2 a^2+2 e m \log \left (f x^2+e\right ) a^2+2 f x^2 \log \left (d \left (f x^2+e\right )^m\right ) a^2+4 b f m n x^2 a-4 b f m x^2 \log \left (c x^n\right ) a+4 b e m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) a+4 b e m n \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) a-2 b e m n \log \left (f x^2+e\right ) a-4 b e m n \log (x) \log \left (f x^2+e\right ) a+4 b e m \log \left (c x^n\right ) \log \left (f x^2+e\right ) a-2 b f n x^2 \log \left (d \left (f x^2+e\right )^m\right ) a+4 b f x^2 \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) a-3 b^2 f m n^2 x^2-2 b^2 f m x^2 \log ^2\left (c x^n\right )+4 b^2 f m n x^2 \log \left (c x^n\right )-2 b^2 e m n^2 \log ^2(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 )+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 b^2 e m n^2 \log ^2(x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )-2 b^2 e m n^2 \log (x) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right )+b^2 e m n^2 \log \left (f x^2+e\right )+2 b^2 e m n^2 \log ^2(x) \log \left (f x^2+e\right )+2 b^2 e m \log ^2\left (c x^n\right ) \log \left (f x^2+e\right )+2 b^2 e m n^2 \log (x) \log \left (f x^2+e\right )-2 b^2 e m n \log \left (c x^n\right ) \log \left (f x^2+e\right )-4 b^2 e m n \log (x) \log \left (c x^n\right ) \log \left (f x^2+e\right )+b^2 f n^2 x^2 \log \left (d \left (f x^2+e\right )^m\right )+2 b^2 f x^2 \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )-2 b^2 f n x^2 \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+2 b e m n \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\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{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )-4 b^2 e m n^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-4 b^2 e m n^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{4 f} \]

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)

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Maple [F]  time = 2.338, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) \, dx \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\left (2 \, b^{2} x^{2} \log \left (x^{n}\right )^{2} - 2 \,{\left (b^{2}{\left (n - 2 \, \log \left (c\right )\right )} - 2 \, a b\right )} x^{2} \log \left (x^{n}\right ) +{\left ({\left (n^{2} - 2 \, n \log \left (c\right ) + 2 \, \log \left (c\right )^{2}\right )} b^{2} - 2 \, a b{\left (n - 2 \, \log \left (c\right )\right )} + 2 \, a^{2}\right )} x^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right ) + \int -\frac{{\left (2 \,{\left (f m - f \log \left (d\right )\right )} a^{2} - 2 \,{\left (f m n - 2 \,{\left (f m - f \log \left (d\right )\right )} \log \left (c\right )\right )} a b +{\left (f m n^{2} - 2 \, f m n \log \left (c\right ) + 2 \,{\left (f m - f \log \left (d\right )\right )} \log \left (c\right )^{2}\right )} b^{2}\right )} x^{3} + 2 \,{\left ({\left (f m - f \log \left (d\right )\right )} b^{2} x^{3} - b^{2} e x \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} - 2 \,{\left (b^{2} e \log \left (c\right )^{2} \log \left (d\right ) + 2 \, a b e \log \left (c\right ) \log \left (d\right ) + a^{2} e \log \left (d\right )\right )} x + 2 \,{\left ({\left (2 \,{\left (f m - f \log \left (d\right )\right )} a b -{\left (f m n - 2 \,{\left (f m - f \log \left (d\right )\right )} \log \left (c\right )\right )} b^{2}\right )} x^{3} - 2 \,{\left (b^{2} e \log \left (c\right ) \log \left (d\right ) + a b e \log \left (d\right )\right )} x\right )} \log \left (x^{n}\right )}{2 \,{\left (f x^{2} + e\right )}}\,{d x} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \end{align*}

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)

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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(x*(a+b*ln(c*x**n))**2*ln(d*(f*x**2+e)**m),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 )}^{2} x \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \end{align*}

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)