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

Optimal. Leaf size=248 \[ -\frac{b f^2 m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{8 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}+\frac{b f^2 m n \log \left (e+f x^2\right )}{16 e^2}-\frac{b f^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{b f^2 m n \log (x)}{8 e^2}-\frac{3 b f m n}{16 e x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.226004, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2454, 2395, 44, 2376, 2301, 2394, 2315} \[ -\frac{b f^2 m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{8 e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 x^4}-\frac{f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{f^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{4 e x^2}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{16 x^4}+\frac{b f^2 m n \log \left (e+f x^2\right )}{16 e^2}-\frac{b f^2 m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 e^2}+\frac{b f^2 m n \log ^2(x)}{4 e^2}-\frac{b f^2 m n \log (x)}{8 e^2}-\frac{3 b f m n}{16 e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -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 2376

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

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 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [C]  time = 0.144964, size = 363, normalized size = 1.46 \[ -\frac{-4 b f^2 m n x^4 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-4 b f^2 m n x^4 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+4 a e^2 \log \left (d \left (e+f x^2\right )^m\right )-4 a f^2 m x^4 \log \left (e+f x^2\right )+4 a e f m x^2+8 a f^2 m x^4 \log (x)+4 b e^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 m x^4 \log \left (c x^n\right ) \log \left (e+f x^2\right )+4 b e f m x^2 \log \left (c x^n\right )+8 b f^2 m x^4 \log (x) \log \left (c x^n\right )+b e^2 n \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 m n x^4 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-4 b f^2 m n x^4 \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-b f^2 m n x^4 \log \left (e+f x^2\right )+4 b f^2 m n x^4 \log (x) \log \left (e+f x^2\right )+3 b e f m n x^2-4 b f^2 m n x^4 \log ^2(x)+2 b f^2 m n x^4 \log (x)}{16 e^2 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*a*e*f*m*x^2 + 3*b*e*f*m*n*x^2 + 8*a*f^2*m*x^4*Log[x] + 2*b*f^2*m*n*x^4*Log[x] - 4*b*f^2*m*n*x^4*Log[x]^2 +
4*b*e*f*m*x^2*Log[c*x^n] + 8*b*f^2*m*x^4*Log[x]*Log[c*x^n] - 4*b*f^2*m*n*x^4*Log[x]*Log[1 - (I*Sqrt[f]*x)/Sqr
t[e]] - 4*b*f^2*m*n*x^4*Log[x]*Log[1 + (I*Sqrt[f]*x)/Sqrt[e]] - 4*a*f^2*m*x^4*Log[e + f*x^2] - b*f^2*m*n*x^4*L
og[e + f*x^2] + 4*b*f^2*m*n*x^4*Log[x]*Log[e + f*x^2] - 4*b*f^2*m*x^4*Log[c*x^n]*Log[e + f*x^2] + 4*a*e^2*Log[
d*(e + f*x^2)^m] + b*e^2*n*Log[d*(e + f*x^2)^m] + 4*b*e^2*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 4*b*f^2*m*n*x^4*Po
lyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]] - 4*b*f^2*m*n*x^4*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(16*e^2*x^4)

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Maple [C]  time = 0.368, size = 2313, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*I/x^4*Pi*a*csgn(I*d*(f*x^2+e)^m)^3+1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x^4*b*csgn(I*c*x^n)^3+1/4*b*f^2*m*n/e
^2*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+1/4*b*f^2*m*n/e^2*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-1/8*b*f^2*
m*n*ln(x)/e^2+1/4*b*f^2*m*n*ln(x)^2/e^2+1/16*b*f^2*m*n*ln(f*x^2+e)/e^2-3/16*b*f*m*n/e/x^2-1/2/e^2*f^2*m*ln(x)*
a+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*(f*x^2+e)^m)*
csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*c)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn
(I*c)*csgn(I*c*x^n)^2-1/8*I/x^4*ln(c)*Pi*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/4/e^2*f^2*m*ln(f*x^2+e)*a+1/8*I
/e^2*f^2*m*ln(f*x^2+e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I/e*f*m/x^2*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I/e*f
*m/x^2*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*x^n
)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/16*Pi^2*csgn(I
*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x^4*b*csgn(I*c*x^n)^3+1/16*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*cs
gn(I*d*(f*x^2+e)^m)/x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I/e^2*f^2*m*ln(f*x^2+e)*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2+(-1/4*b/x^4*ln(x^n)-1/16*(-2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*I*b*Pi*csgn(I*c)*csgn(I
*c*x^n)^2+2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*c*x^n)^3+4*b*ln(c)+b*n+4*a)/x^4)*ln((f*x^2+e)^m
)+1/4*b*f^2*m*n/e^2*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+1/4*b*f^2*m*n/e^2*ln(x)*ln((f*x+(-e*f)^(1/2))/(
-e*f)^(1/2))-1/4*b*f^2*m*n*ln(x)/e^2*ln(f*x^2+e)+1/32*I/x^4*Pi*b*n*csgn(I*d*(f*x^2+e)^m)^3-1/8*I/x^4*Pi*a*csgn
(I*d)*csgn(I*d*(f*x^2+e)^m)^2-1/4*I/e^2*f^2*m*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/4*I/e^2*f^2*m*ln(x)*b*Pi*
csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I/e^2*f^2*m*ln(x)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/16*Pi^2*csgn(I*d)
*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)/x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csg
n(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/16*Pi^2*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*
d*(f*x^2+e)^m)/x^4*b*csgn(I*c)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)+1/8*I/x^4*ln(c)*Pi*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)+1/8*I/x^4*Pi*l
n(d)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I*Pi*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)*b/x^4*
ln(x^n)+1/32*I/x^4*Pi*b*n*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)+1/8*I/e*f*m/x^2*b*Pi*csgn(I*c*x^
n)^3-1/8*I/e^2*f^2*m*ln(f*x^2+e)*b*Pi*csgn(I*c*x^n)^3-1/4/e*f*m/x^2*b*ln(c)-1/4/x^4*ln(c)*ln(d)*b-1/16/x^4*ln(
d)*b*n+1/4*I/e^2*f^2*m*ln(x)*b*Pi*csgn(I*c*x^n)^3-1/2/e^2*f^2*m*ln(x)*b*ln(c)-1/2*m*f^2*b*ln(x^n)/e^2*ln(x)-1/
4*m*f*b*ln(x^n)/e/x^2-1/4/e*f*m/x^2*a+1/4*m*f^2*b*ln(x^n)/e^2*ln(f*x^2+e)-1/4/x^4*ln(d)*a-1/4*ln(d)*b/x^4*ln(x
^n)-1/8*I/x^4*Pi*a*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2+1/8*I/x^4*ln(c)*Pi*b*csgn(I*d*(f*x^2+e)^m)^3-1/
16*Pi^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2/x^4*b*csgn(I*c*x^n)^3-1/16*Pi^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+
e)^m)^2/x^4*b*csgn(I*c*x^n)^3-1/16*Pi^2*csgn(I*d*(f*x^2+e)^m)^3/x^4*b*csgn(I*c)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn
(I*d*(f*x^2+e)^m)^3/x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4/e^2*f^2*m*ln(f*x^2+e)*b*ln(c)+1/8*I/x^4*Pi*a*csgn(I*
d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-1/8*I*Pi*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2*b/x^4*ln(x^n)-1/8*I*Pi
*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2*b/x^4*ln(x^n)-1/32*I/x^4*Pi*b*n*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2
-1/32*I/x^4*Pi*b*n*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/8*I/x^4*ln(c)*Pi*b*csgn(I*(f*x^2+e)^m)*csgn(I
*d*(f*x^2+e)^m)^2-1/8*I/x^4*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I/x^4*Pi*ln(d)*b*csgn(I*x^n)*csgn(I*c*x^n
)^2+1/8*I*Pi*csgn(I*d*(f*x^2+e)^m)^3*b/x^4*ln(x^n)+1/8*I/x^4*Pi*ln(d)*b*csgn(I*c*x^n)^3-1/8*I/e^2*f^2*m*ln(f*x
^2+e)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I/e*f*m/x^2*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b{\left (n + 4 \, \log \left (c\right )\right )} + 4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} \log \left ({\left (f x^{2} + e\right )}^{m}\right )}{16 \, x^{4}} + \int \frac{8 \, b e \log \left (c\right ) \log \left (d\right ) +{\left (4 \,{\left (f m + 2 \, f \log \left (d\right )\right )} a +{\left (f m n + 4 \,{\left (f m + 2 \, f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{2} + 8 \, a e \log \left (d\right ) + 4 \,{\left ({\left (f m + 2 \, f \log \left (d\right )\right )} b x^{2} + 2 \, b e \log \left (d\right )\right )} \log \left (x^{n}\right )}{8 \,{\left (f x^{7} + e x^{5}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)*log((f*x^2 + e)^m*d)/x^5, 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))*ln(d*(f*x**2+e)**m)/x**5,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 )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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