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

Optimal. Leaf size=276 \[ \frac{b f m n \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{b^2 f m n^2 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right )}{4 e}+\frac{b^2 f m n^2 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right )}{4 e}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{b f m n \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{f m \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{b^2 f m n^2 \log \left (e+f x^2\right )}{4 e}+\frac{b^2 f m n^2 \log (x)}{2 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.330023, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {2305, 2304, 2378, 266, 36, 29, 31, 2345, 2391, 2374, 6589} \[ \frac{b f m n \text{PolyLog}\left (2,-\frac{e}{f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{b^2 f m n^2 \text{PolyLog}\left (2,-\frac{e}{f x^2}\right )}{4 e}+\frac{b^2 f m n^2 \text{PolyLog}\left (3,-\frac{e}{f x^2}\right )}{4 e}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}-\frac{b f m n \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{f m \log \left (\frac{e}{f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{b^2 n^2 \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{b^2 f m n^2 \log \left (e+f x^2\right )}{4 e}+\frac{b^2 f m n^2 \log (x)}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.463225, size = 946, normalized size = 3.43 \[ -\frac{-4 b^2 f m n^2 x^2 \log ^3(x)+6 b^2 f m n^2 x^2 \log ^2(x)+12 a b f m n x^2 \log ^2(x)+12 b^2 f m n x^2 \log \left (c x^n\right ) \log ^2(x)-6 b^2 f m n^2 x^2 \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \log ^2(x)-6 b^2 f m n^2 x^2 \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) \log ^2(x)+6 b^2 f m n^2 x^2 \log \left (f x^2+e\right ) \log ^2(x)-6 b^2 f m n^2 x^2 \log (x)-12 a^2 f m x^2 \log (x)-12 a b f m n x^2 \log (x)-12 b^2 f m x^2 \log ^2\left (c x^n\right ) \log (x)-24 a b f m x^2 \log \left (c x^n\right ) \log (x)-12 b^2 f m n x^2 \log \left (c x^n\right ) \log (x)+6 b^2 f m n^2 x^2 \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \log (x)+12 a b f m n x^2 \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \log (x)+12 b^2 f m n x^2 \log \left (c x^n\right ) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) \log (x)+6 b^2 f m n^2 x^2 \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) \log (x)+12 a b f m n x^2 \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) \log (x)+12 b^2 f m n x^2 \log \left (c x^n\right ) \log \left (\frac{i \sqrt{f} x}{\sqrt{e}}+1\right ) \log (x)-6 b^2 f m n^2 x^2 \log \left (f x^2+e\right ) \log (x)-12 a b f m n x^2 \log \left (f x^2+e\right ) \log (x)-12 b^2 f m n x^2 \log \left (c x^n\right ) \log \left (f x^2+e\right ) \log (x)+3 b^2 f m n^2 x^2 \log \left (f x^2+e\right )+6 a^2 f m x^2 \log \left (f x^2+e\right )+6 a b f m n x^2 \log \left (f x^2+e\right )+6 b^2 f m x^2 \log ^2\left (c x^n\right ) \log \left (f x^2+e\right )+12 a b f m x^2 \log \left (c x^n\right ) \log \left (f x^2+e\right )+6 b^2 f m n x^2 \log \left (c x^n\right ) \log \left (f x^2+e\right )+3 b^2 e n^2 \log \left (d \left (f x^2+e\right )^m\right )+6 b^2 e \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+6 a^2 e \log \left (d \left (f x^2+e\right )^m\right )+6 a b e n \log \left (d \left (f x^2+e\right )^m\right )+12 a b e \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+6 b^2 e n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right )+6 b f m n x^2 \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 )+6 b f m n x^2 \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 )-12 b^2 f m n^2 x^2 \text{PolyLog}\left (3,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-12 b^2 f m n^2 x^2 \text{PolyLog}\left (3,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{12 e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-12*a^2*f*m*x^2*Log[x] - 12*a*b*f*m*n*x^2*Log[x] - 6*b^2*f*m*n^2*x^2*Log[x] + 12*a*b*f*m*n*x^2*Log[x]^2 + 6*
b^2*f*m*n^2*x^2*Log[x]^2 - 4*b^2*f*m*n^2*x^2*Log[x]^3 - 24*a*b*f*m*x^2*Log[x]*Log[c*x^n] - 12*b^2*f*m*n*x^2*Lo
g[x]*Log[c*x^n] + 12*b^2*f*m*n*x^2*Log[x]^2*Log[c*x^n] - 12*b^2*f*m*x^2*Log[x]*Log[c*x^n]^2 + 12*a*b*f*m*n*x^2
*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^2*f*m*n^2*x^2*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] - 6*b^2*f*m*n
^2*x^2*Log[x]^2*Log[1 - (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 - (I*Sqrt[f]*x)/Sqrt
[e]] + 12*a*b*f*m*n*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*b^2*f*m*n^2*x^2*Log[x]*Log[1 + (I*Sqrt[f]*x)
/Sqrt[e]] - 6*b^2*f*m*n^2*x^2*Log[x]^2*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 12*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log
[1 + (I*Sqrt[f]*x)/Sqrt[e]] + 6*a^2*f*m*x^2*Log[e + f*x^2] + 6*a*b*f*m*n*x^2*Log[e + f*x^2] + 3*b^2*f*m*n^2*x^
2*Log[e + f*x^2] - 12*a*b*f*m*n*x^2*Log[x]*Log[e + f*x^2] - 6*b^2*f*m*n^2*x^2*Log[x]*Log[e + f*x^2] + 6*b^2*f*
m*n^2*x^2*Log[x]^2*Log[e + f*x^2] + 12*a*b*f*m*x^2*Log[c*x^n]*Log[e + f*x^2] + 6*b^2*f*m*n*x^2*Log[c*x^n]*Log[
e + f*x^2] - 12*b^2*f*m*n*x^2*Log[x]*Log[c*x^n]*Log[e + f*x^2] + 6*b^2*f*m*x^2*Log[c*x^n]^2*Log[e + f*x^2] + 6
*a^2*e*Log[d*(e + f*x^2)^m] + 6*a*b*e*n*Log[d*(e + f*x^2)^m] + 3*b^2*e*n^2*Log[d*(e + f*x^2)^m] + 12*a*b*e*Log
[c*x^n]*Log[d*(e + f*x^2)^m] + 6*b^2*e*n*Log[c*x^n]*Log[d*(e + f*x^2)^m] + 6*b^2*e*Log[c*x^n]^2*Log[d*(e + f*x
^2)^m] + 6*b*f*m*n*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] + 6*b*f*m*n*x^2*(2*a
+ b*n + 2*b*Log[c*x^n])*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]] - 12*b^2*f*m*n^2*x^2*PolyLog[3, ((-I)*Sqrt[f]*x)/Sqr
t[e]] - 12*b^2*f*m*n^2*x^2*PolyLog[3, (I*Sqrt[f]*x)/Sqrt[e]])/(12*e*x^2)

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Maple [F]  time = 1.664, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (2 \, b^{2} \log \left (x^{n}\right )^{2} +{\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} + 2 \,{\left (b^{2}{\left (n + 2 \, \log \left (c\right )\right )} + 2 \, a b\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right )}{4 \, x^{2}} + \int \frac{2 \, b^{2} e \log \left (c\right )^{2} \log \left (d\right ) + 4 \, a b e \log \left (c\right ) \log \left (d\right ) + 2 \, a^{2} e \log \left (d\right ) +{\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^{2} + 2 \,{\left ({\left (f m + f \log \left (d\right )\right )} b^{2} x^{2} + b^{2} e \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (2 \, b^{2} e \log \left (c\right ) \log \left (d\right ) + 2 \, a b e \log \left (d\right ) +{\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^{2}\right )} \log \left (x^{n}\right )}{2 \,{\left (f x^{5} + e x^{3}\right )}}\,{d x} \end{align*}

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^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{3}}, x\right ) \end{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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^{3}}\,{d x} \end{align*}

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^3,x, algorithm="giac")

[Out]

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