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

Optimal. Leaf size=150 \[ \frac{2 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right )}{m^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

[Out]

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

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Rubi [A]  time = 0.248844, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ \frac{2 b n r \text{PolyLog}\left (3,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m^2}-\frac{r \text{PolyLog}\left (2,-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{f x^m}{e}\right )}{m^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \log \left (\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2375

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

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 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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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^m\right )^r\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{(f m r) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^m} \, dx}{3 b n}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}+r \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x^m}{e}\right )}{x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{(2 b n r) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{2 b n r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{\left (2 b^2 n^2 r\right ) \int \frac{\text{Li}_3\left (-\frac{f x^m}{e}\right )}{x} \, dx}{m^2}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^m\right )^r\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x^m}{e}\right )}{3 b n}-\frac{r \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x^m}{e}\right )}{m}+\frac{2 b n r \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x^m}{e}\right )}{m^2}-\frac{2 b^2 n^2 r \text{Li}_4\left (-\frac{f x^m}{e}\right )}{m^3}\\ \end{align*}

Mathematica [B]  time = 0.360166, size = 741, normalized size = 4.94 \[ \frac{b n r \log (x) \text{PolyLog}\left (2,-\frac{e x^{-m}}{f}\right ) \left (2 \left (a+b \log \left (c x^n\right )\right )-b n \log (x)\right )}{m}+\frac{r \text{PolyLog}\left (2,\frac{f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2}{m}+\frac{2 a b n r \text{PolyLog}\left (3,-\frac{e x^{-m}}{f}\right )}{m^2}+\frac{2 b^2 n r \log \left (c x^n\right ) \text{PolyLog}\left (3,-\frac{e x^{-m}}{f}\right )}{m^2}+\frac{2 b^2 n^2 r \text{PolyLog}\left (4,-\frac{e x^{-m}}{f}\right )}{m^3}+a^2 \log (x) \log \left (d \left (e+f x^m\right )^r\right )-a^2 r \log (x) \log \left (e+f x^m\right )+\frac{a^2 r \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+2 a b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-2 a b r \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac{2 a b r \log \left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-a b n \log ^2(x) \log \left (d \left (e+f x^m\right )^r\right )-a b n r \log ^2(x) \log \left (\frac{e x^{-m}}{f}+1\right )+2 a b n r \log ^2(x) \log \left (e+f x^m\right )-\frac{2 a b n r \log (x) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{3} a b m n r \log ^3(x)-b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )+b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (\frac{e x^{-m}}{f}+1\right )+2 b^2 n r \log ^2(x) \log \left (c x^n\right ) \log \left (e+f x^m\right )-b^2 r \log (x) \log ^2\left (c x^n\right ) \log \left (e+f x^m\right )+\frac{b^2 r \log ^2\left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{2 b^2 n r \log (x) \log \left (c x^n\right ) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac{1}{3} b^2 m n r \log ^3(x) \log \left (c x^n\right )+\frac{1}{3} b^2 n^2 \log ^3(x) \log \left (d \left (e+f x^m\right )^r\right )+\frac{2}{3} b^2 n^2 r \log ^3(x) \log \left (\frac{e x^{-m}}{f}+1\right )-b^2 n^2 r \log ^3(x) \log \left (e+f x^m\right )+\frac{b^2 n^2 r \log ^2(x) \log \left (-\frac{f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}+\frac{1}{4} b^2 m n^2 r \log ^4(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( e+f{x}^{m} \right ) ^{r} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (b^{2} n^{2} \log \left (x\right )^{3} + 3 \, b^{2} \log \left (x\right ) \log \left (x^{n}\right )^{2} - 3 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )^{2} - 3 \,{\left (b^{2} n \log \left (x\right )^{2} - 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x\right )\right )} \log \left (x^{n}\right ) + 3 \,{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (x\right )\right )} \log \left ({\left (f x^{m} + e\right )}^{r}\right ) - \int -\frac{3 \, b^{2} e \log \left (c\right )^{2} \log \left (d\right ) + 6 \, a b e \log \left (c\right ) \log \left (d\right ) + 3 \, a^{2} e \log \left (d\right ) + 3 \,{\left (b^{2} e \log \left (d\right ) -{\left (b^{2} f m r \log \left (x\right ) - b^{2} f \log \left (d\right )\right )} x^{m}\right )} \log \left (x^{n}\right )^{2} -{\left (b^{2} f m n^{2} r \log \left (x\right )^{3} - 3 \, b^{2} f \log \left (c\right )^{2} \log \left (d\right ) - 6 \, a b f \log \left (c\right ) \log \left (d\right ) - 3 \, a^{2} f \log \left (d\right ) - 3 \,{\left (b^{2} f m n r \log \left (c\right ) + a b f m n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} f m r \log \left (c\right )^{2} + 2 \, a b f m r \log \left (c\right ) + a^{2} f m r\right )} \log \left (x\right )\right )} x^{m} + 3 \,{\left (2 \, b^{2} e \log \left (c\right ) \log \left (d\right ) + 2 \, a b e \log \left (d\right ) +{\left (b^{2} f m n r \log \left (x\right )^{2} + 2 \, b^{2} f \log \left (c\right ) \log \left (d\right ) + 2 \, a b f \log \left (d\right ) - 2 \,{\left (b^{2} f m r \log \left (c\right ) + a b f m r\right )} \log \left (x\right )\right )} x^{m}\right )} \log \left (x^{n}\right )}{3 \,{\left (f x x^{m} + e x\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*(e+f*x^m)^r)/x,x, algorithm="maxima")

[Out]

1/3*(b^2*n^2*log(x)^3 + 3*b^2*log(x)*log(x^n)^2 - 3*(b^2*n*log(c) + a*b*n)*log(x)^2 - 3*(b^2*n*log(x)^2 - 2*(b
^2*log(c) + a*b)*log(x))*log(x^n) + 3*(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*log(x))*log((f*x^m + e)^r) - integra
te(-1/3*(3*b^2*e*log(c)^2*log(d) + 6*a*b*e*log(c)*log(d) + 3*a^2*e*log(d) + 3*(b^2*e*log(d) - (b^2*f*m*r*log(x
) - b^2*f*log(d))*x^m)*log(x^n)^2 - (b^2*f*m*n^2*r*log(x)^3 - 3*b^2*f*log(c)^2*log(d) - 6*a*b*f*log(c)*log(d)
- 3*a^2*f*log(d) - 3*(b^2*f*m*n*r*log(c) + a*b*f*m*n*r)*log(x)^2 + 3*(b^2*f*m*r*log(c)^2 + 2*a*b*f*m*r*log(c)
+ a^2*f*m*r)*log(x))*x^m + 3*(2*b^2*e*log(c)*log(d) + 2*a*b*e*log(d) + (b^2*f*m*n*r*log(x)^2 + 2*b^2*f*log(c)*
log(d) + 2*a*b*f*log(d) - 2*(b^2*f*m*r*log(c) + a*b*f*m*r)*log(x))*x^m)*log(x^n))/(f*x*x^m + e*x), x)

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Fricas [C]  time = 1.10778, size = 995, normalized size = 6.63 \begin{align*} \frac{b^{2} m^{3} n^{2} \log \left (d\right ) \log \left (x\right )^{3} - 6 \, b^{2} n^{2} r{\rm polylog}\left (4, -\frac{f x^{m}}{e}\right ) + 3 \,{\left (b^{2} m^{3} n \log \left (c\right ) + a b m^{3} n\right )} \log \left (d\right ) \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} \log \left (c\right )^{2} + 2 \, a b m^{3} \log \left (c\right ) + a^{2} m^{3}\right )} \log \left (d\right ) \log \left (x\right ) - 3 \,{\left (b^{2} m^{2} n^{2} r \log \left (x\right )^{2} + b^{2} m^{2} r \log \left (c\right )^{2} + 2 \, a b m^{2} r \log \left (c\right ) + a^{2} m^{2} r + 2 \,{\left (b^{2} m^{2} n r \log \left (c\right ) + a b m^{2} n r\right )} \log \left (x\right )\right )}{\rm Li}_2\left (-\frac{f x^{m} + e}{e} + 1\right ) +{\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \,{\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) -{\left (b^{2} m^{3} n^{2} r \log \left (x\right )^{3} + 3 \,{\left (b^{2} m^{3} n r \log \left (c\right ) + a b m^{3} n r\right )} \log \left (x\right )^{2} + 3 \,{\left (b^{2} m^{3} r \log \left (c\right )^{2} + 2 \, a b m^{3} r \log \left (c\right ) + a^{2} m^{3} r\right )} \log \left (x\right )\right )} \log \left (\frac{f x^{m} + e}{e}\right ) + 6 \,{\left (b^{2} m n^{2} r \log \left (x\right ) + b^{2} m n r \log \left (c\right ) + a b m n r\right )}{\rm polylog}\left (3, -\frac{f x^{m}}{e}\right )}{3 \, m^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^2*m^3*n^2*log(d)*log(x)^3 - 6*b^2*n^2*r*polylog(4, -f*x^m/e) + 3*(b^2*m^3*n*log(c) + a*b*m^3*n)*log(d)*
log(x)^2 + 3*(b^2*m^3*log(c)^2 + 2*a*b*m^3*log(c) + a^2*m^3)*log(d)*log(x) - 3*(b^2*m^2*n^2*r*log(x)^2 + b^2*m
^2*r*log(c)^2 + 2*a*b*m^2*r*log(c) + a^2*m^2*r + 2*(b^2*m^2*n*r*log(c) + a*b*m^2*n*r)*log(x))*dilog(-(f*x^m +
e)/e + 1) + (b^2*m^3*n^2*r*log(x)^3 + 3*(b^2*m^3*n*r*log(c) + a*b*m^3*n*r)*log(x)^2 + 3*(b^2*m^3*r*log(c)^2 +
2*a*b*m^3*r*log(c) + a^2*m^3*r)*log(x))*log(f*x^m + e) - (b^2*m^3*n^2*r*log(x)^3 + 3*(b^2*m^3*n*r*log(c) + a*b
*m^3*n*r)*log(x)^2 + 3*(b^2*m^3*r*log(c)^2 + 2*a*b*m^3*r*log(c) + a^2*m^3*r)*log(x))*log((f*x^m + e)/e) + 6*(b
^2*m*n^2*r*log(x) + b^2*m*n*r*log(c) + a*b*m*n*r)*polylog(3, -f*x^m/e))/m^3

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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*(e+f*x**m)**r)/x,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^{m} + e\right )}^{r} d\right )}{x}\,{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*(e+f*x^m)^r)/x,x, algorithm="giac")

[Out]

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