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

Optimal. Leaf size=194 \[ -\frac{i b \sqrt{e} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{i b \sqrt{e} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}-2 m x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 b \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+4 b m n x \]

[Out]

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

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Rubi [A]  time = 0.115724, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2448, 321, 205, 2370, 4848, 2391} \[ -\frac{i b \sqrt{e} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{i b \sqrt{e} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac{2 \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{f}}-2 m x \left (a+b \log \left (c x^n\right )\right )-b n x \log \left (d \left (e+f x^2\right )^m\right )-\frac{2 b \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+4 b m n x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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]

Rubi steps

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

Mathematica [A]  time = 0.0771769, size = 332, normalized size = 1.71 \[ \frac{-i b \sqrt{e} m n \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right )+i b \sqrt{e} m n \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right )+a \sqrt{f} x \log \left (d \left (e+f x^2\right )^m\right )+2 a \sqrt{e} m \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-2 a \sqrt{f} m x+b \sqrt{f} x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b \sqrt{e} m \log \left (c x^n\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-2 b \sqrt{f} m x \log \left (c x^n\right )-b \sqrt{f} n x \log \left (d \left (e+f x^2\right )^m\right )+i b \sqrt{e} m n \log (x) \log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )-i b \sqrt{e} m n \log (x) \log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )-2 b \sqrt{e} m n \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )-2 b \sqrt{e} m n \log (x) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )+4 b \sqrt{f} m n x}{\sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.158, size = 2001, normalized size = 10.3 \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)

[Out]

1/2*I*Pi*x*a*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^
2+e)^m)^2+2*m*b*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*ln(x^n)+1/2*I*Pi*x*a*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2
+e)^m)^2-1/2*I*ln(x^n)*Pi*x*b*csgn(I*d*(f*x^2+e)^m)^3-1/2*I*ln(c)*Pi*x*b*csgn(I*d*(f*x^2+e)^m)^3+1/2*I*Pi*x*b*
n*csgn(I*d*(f*x^2+e)^m)^3+1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/4*Pi^2*x*b*csgn(I*x
^n)*csgn(I*c*x^n)^2*csgn(I*d*(f*x^2+e)^m)^3+I*m*x*b*Pi*csgn(I*c*x^n)^3+(b*x*ln(x^n)+1/2*(-I*b*Pi*csgn(I*c)*csg
n(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)-2*b*n+2*a)*x)*ln((f*x^2+e)^m)+4*b*m*n*x+ln(d)*a*x-2*m*b*ln(x^n)*x+ln(x^n)*ln(d)*x*b-2*m*ln(c)*b*
x+ln(c)*ln(d)*b*x-ln(d)*b*n*x+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*d*(f*x^2+e)^m)^3-2*m*e/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*b*n+m*b*n*e/(-e*f)^(1/2)*dilog((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-m*b*n*e/(-e*f)^(1/2)
*dilog((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+2*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*ln(c)-2*m*b*e/(e*f)^(1/2)*
arctan(x*f/(e*f)^(1/2))*n*ln(x)+m*b*n*e/(-e*f)^(1/2)*ln(x)*ln((-f*x+(-e*f)^(1/2))/(-e*f)^(1/2))-m*b*n*e/(-e*f)
^(1/2)*ln(x)*ln((f*x+(-e*f)^(1/2))/(-e*f)^(1/2))+I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn
(I*c*x^n)^2-1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^
m)+I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*x*b*n*csgn(I*d)*csgn(I*
(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-I*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c*x^n)^3-1/2*I*ln(x^n
)*Pi*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*
csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*(f*x^2+e)^m)*csgn(I*
d*(f*x^2+e)^m)^2+I*m*x*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*
d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)+1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(I*(f*x^2+
e)^m)*csgn(I*d*(f*x^2+e)^m)-1/2*I*ln(c)*Pi*x*b*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-2*a*m*x-I*m
*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*Pi^2*x*b*csgn(I*c*x^n)^3*c
sgn(I*d*(f*x^2+e)^m)^3-1/2*I*Pi*x*a*csgn(I*d*(f*x^2+e)^m)^3+2*a*m*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))-1/4*Pi
^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*d*(f*x^2+e)^m)^3-1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*(f*x^2+e)^m)*csgn(I*d*
(f*x^2+e)^m)^2+1/2*I*ln(c)*Pi*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*Pi*x*b*n*csgn(I*(f*x^2+e)^
m)*csgn(I*d*(f*x^2+e)^m)^2-1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*Pi
*ln(d)*b*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*ln(c)*Pi*x*b*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*Pi*x
*b*n*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/2*I*ln(x^n)*Pi*x*b*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)^2-1/2*I*
Pi*x*a*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-I*m*x*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*m*x*b*Pi*csg
n(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*d)*csgn(I*(f*x^2+e)^m)*csgn(I*d*(f*x^2+e)^m)-1/4*
Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(I*d*(f*x^2+e)^m)^2+1/2*I*ln(x^n)*Pi*x*b*csgn(I*d)*csgn(I*d*(
f*x^2+e)^m)^2-1/2*I*Pi*ln(d)*b*x*csgn(I*c*x^n)^3+1/2*I*Pi*ln(d)*b*x*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*ln(d)
*b*x*csgn(I*c)*csgn(I*c*x^n)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

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

[Out]

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