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

Optimal. Leaf size=195 \[ \frac{b f m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{4 e}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{f m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{b f m n \log \left (e+f x^2\right )}{4 e}+\frac{b f m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{4 e}-\frac{b f m n \log ^2(x)}{2 e}+\frac{b f m n \log (x)}{2 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.181568, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2454, 2395, 36, 29, 31, 2376, 2301, 2394, 2315} \[ \frac{b f m n \text{PolyLog}\left (2,\frac{f x^2}{e}+1\right )}{4 e}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 x^2}+\frac{f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{f m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b n \log \left (d \left (e+f x^2\right )^m\right )}{4 x^2}-\frac{b f m n \log \left (e+f x^2\right )}{4 e}+\frac{b f m n \log \left (-\frac{f x^2}{e}\right ) \log \left (e+f x^2\right )}{4 e}-\frac{b f m n \log ^2(x)}{2 e}+\frac{b f m n \log (x)}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.126418, size = 298, normalized size = 1.53 \[ -\frac{2 b f m n x^2 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b f m n x^2 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 a e \log \left (d \left (e+f x^2\right )^m\right )+2 a f m x^2 \log \left (e+f x^2\right )-4 a f m x^2 \log (x)+2 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b f m x^2 \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 b f m x^2 \log (x) \log \left (c x^n\right )+b e n \log \left (d \left (e+f x^2\right )^m\right )+2 b f m n x^2 \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+2 b f m n x^2 \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )+b f m n x^2 \log \left (e+f x^2\right )-2 b f m n x^2 \log (x) \log \left (e+f x^2\right )+2 b f m n x^2 \log ^2(x)-2 b f m n x^2 \log (x)}{4 e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.314, size = 2101, normalized size = 10.8 \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^3,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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^{3}}, 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^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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**3,x)

[Out]

Timed out

________________________________________________________________________________________

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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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