3.3.32 \(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x) (h+i x)} \, dx\) [232]

Optimal. Leaf size=372 \[ \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]

[Out]

(a+b*ln(c*(e*x+d)^n))^3*ln(e*(g*x+f)/(-d*g+e*f))/(-f*i+g*h)-(a+b*ln(c*(e*x+d)^n))^3*ln(e*(i*x+h)/(-d*i+e*h))/(
-f*i+g*h)+3*b*n*(a+b*ln(c*(e*x+d)^n))^2*polylog(2,-g*(e*x+d)/(-d*g+e*f))/(-f*i+g*h)-3*b*n*(a+b*ln(c*(e*x+d)^n)
)^2*polylog(2,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)-6*b^2*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,-g*(e*x+d)/(-d*g+e*f
))/(-f*i+g*h)+6*b^2*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)+6*b^3*n^3*polylog(4,
-g*(e*x+d)/(-d*g+e*f))/(-f*i+g*h)-6*b^3*n^3*polylog(4,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2465, 2443, 2481, 2421, 2430, 6724} \begin {gather*} -\frac {6 b^2 n^2 \text {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {6 b^2 n^2 \text {PolyLog}\left (3,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {3 b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}-\frac {3 b n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g h-f i}+\frac {6 b^3 n^3 \text {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {6 b^3 n^3 \text {PolyLog}\left (4,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g h-f i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) - ((a + b*Log[c*(d + e*x)^n])^3*Log[
(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i) + (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f
- d*g))])/(g*h - f*i) - (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f
*i) - (6*b^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) + (6*b^2*n^2
*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i) + (6*b^3*n^3*PolyLog[4, -((g
*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (6*b^3*n^3*PolyLog[4, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2443

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

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

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 (d+e x)^n\right )\right )^3}{(h+232 x) (f+g x)} \, dx &=\int \left (\frac {232 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(232 f-g h) (h+232 x)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(232 f-g h) (f+g x)}\right ) \, dx\\ &=\frac {232 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{h+232 x} \, dx}{232 f-g h}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {(3 b e n) \int \frac {\log \left (\frac {e (h+232 x)}{-232 d+e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx}{232 f-g h}+\frac {(3 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {-232 d+e h}{e}+\frac {232 x}{e}\right )}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}+\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {\left (6 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}-\frac {\left (6 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {232 x}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}-\frac {\left (6 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}+\frac {\left (6 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {232 x}{-232 d+e h}\right )}{x} \, dx,x,d+e x\right )}{232 f-g h}\\ &=\frac {\log \left (-\frac {e (h+232 x)}{232 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{232 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{232 f-g h}-\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}+\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}-\frac {6 b^3 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{232 f-g h}+\frac {6 b^3 n^3 \text {Li}_4\left (\frac {232 (d+e x)}{232 d-e h}\right )}{232 f-g h}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.25, size = 599, normalized size = 1.61 \begin {gather*} \frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)-\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (h+i x)+3 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+\text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-\text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+\log (d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-\log (d+e x) \text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )-\text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )+\text {Li}_3\left (\frac {i (d+e x)}{-e h+d i}\right )\right )+b^3 n^3 \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\log ^3(d+e x) \log \left (\frac {e (h+i x)}{e h-d i}\right )+3 \log ^2(d+e x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-3 \log ^2(d+e x) \text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )-6 \log (d+e x) \text {Li}_3\left (\frac {g (d+e x)}{-e f+d g}\right )+6 \log (d+e x) \text {Li}_3\left (\frac {i (d+e x)}{-e h+d i}\right )+6 \text {Li}_4\left (\frac {g (d+e x)}{-e f+d g}\right )-6 \text {Li}_4\left (\frac {i (d+e x)}{-e h+d i}\right )\right )}{g h-f i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*Log[f + g*x] - (a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^
3*Log[h + i*x] + 3*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(Log[d + e*x]*(Log[(e*(f + g*x))/(e*f -
d*g)] - Log[(e*(h + i*x))/(e*h - d*i)]) + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - PolyLog[2, (i*(d + e*x))
/(-(e*h) + d*i)]) + 6*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((Log[d + e*x]^2*Log[(e*(f + g*x))
/(e*f - d*g)])/2 - (Log[d + e*x]^2*Log[(e*(h + i*x))/(e*h - d*i)])/2 + Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(
-(e*f) + d*g)] - Log[d + e*x]*PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)] - PolyLog[3, (g*(d + e*x))/(-(e*f) + d*
g)] + PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)]) + b^3*n^3*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] - Log
[d + e*x]^3*Log[(e*(h + i*x))/(e*h - d*i)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 3*Log
[d + e*x]^2*PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]
+ 6*Log[d + e*x]*PolyLog[3, (i*(d + e*x))/(-(e*h) + d*i)] + 6*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)] - 6*Po
lyLog[4, (i*(d + e*x))/(-(e*h) + d*i)]))/(g*h - f*i)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.77, size = 21696, normalized size = 58.32

method	result	size
risch	\(\text {Expression too large to display}\)	\(21696\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="maxima")

[Out]

a^3*(log(g*x + f)/(g*h - I*f) - log(h + I*x)/(g*h - I*f)) - integrate((I*b^3*log((x*e + d)^n)^3 + I*b^3*log(c)
^3 + 3*I*a*b^2*log(c)^2 + 3*I*a^2*b*log(c) - 3*(-I*b^3*log(c) - I*a*b^2)*log((x*e + d)^n)^2 - 3*(-I*b^3*log(c)
^2 - 2*I*a*b^2*log(c) - I*a^2*b)*log((x*e + d)^n))/(g*x^2 - I*f*h + (-I*g*h + f)*x), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="fricas")

[Out]

integral((-I*b^3*n^3*log(x*e + d)^3 - I*b^3*log(c)^3 - 3*I*a*b^2*log(c)^2 - 3*I*a^2*b*log(c) - I*a^3 - 3*(I*b^
3*n^2*log(c) + I*a*b^2*n^2)*log(x*e + d)^2 - 3*(I*b^3*n*log(c)^2 + 2*I*a*b^2*n*log(c) + I*a^2*b*n)*log(x*e + d
))/(g*x^2 - I*f*h + (-I*g*h + f)*x), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**3/(g*x+f)/(i*x+h),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**3/((f + g*x)*(h + i*x)), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3/(g*x+f)/(i*x+h),x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)^3/((g*x + f)*(h + I*x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{\left (f+g\,x\right )\,\left (h+i\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^3/((f + g*x)*(h + i*x)),x)

[Out]

int((a + b*log(c*(d + e*x)^n))^3/((f + g*x)*(h + i*x)), x)

________________________________________________________________________________________