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

Optimal. Leaf size=295 \[ -\frac{3 b^2 n^2 (d g+e f) \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}+\frac{3 b^3 n^3 (e f-d g) \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^2 e^2}+\frac{3 b^3 n^3 (d g+e f) \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^2 e^2}+\frac{3 b^2 n^2 (e f-d g) \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac{3 b n (d g+e f) \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 (e f-d g)}-\frac{3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac{(f+g x)^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (d+e x)^2 (e f-d g)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.624214, antiderivative size = 408, normalized size of antiderivative = 1.38, number of steps used = 17, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2357, 2319, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac{3 b^2 n^2 (e f-d g) \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac{6 b^2 g n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d e^2}+\frac{3 b^3 n^3 (e f-d g) \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^2 e^2}+\frac{3 b^3 n^3 (e f-d g) \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^2 e^2}+\frac{6 b^3 g n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d e^2}+\frac{3 b^2 n^2 (e f-d g) \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e^2}-\frac{3 b n (e f-d g) \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e^2}+\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac{3 b n x (e f-d g) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}-\frac{3 b g n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d e^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*(e*f - d*g)*n*x*(a + b*Log[c*x^n])^2)/(2*d^2*e*(d + e*x)) + ((e*f - d*g)*(a + b*Log[c*x^n])^3)/(2*d^2*e^
2) - ((e*f - d*g)*(a + b*Log[c*x^n])^3)/(2*e^2*(d + e*x)^2) + (g*x*(a + b*Log[c*x^n])^3)/(d*e*(d + e*x)) + (3*
b^2*(e*f - d*g)*n^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(d^2*e^2) - (3*b*g*n*(a + b*Log[c*x^n])^2*Log[1 + (e*
x)/d])/(d*e^2) - (3*b*(e*f - d*g)*n*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/(2*d^2*e^2) + (3*b^3*(e*f - d*g)*n^
3*PolyLog[2, -((e*x)/d)])/(d^2*e^2) - (6*b^2*g*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/(d*e^2) - (3*b^2
*(e*f - d*g)*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/(d^2*e^2) + (6*b^3*g*n^3*PolyLog[3, -((e*x)/d)])/(
d*e^2) + (3*b^3*(e*f - d*g)*n^3*PolyLog[3, -((e*x)/d)])/(d^2*e^2)

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

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

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 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]

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

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 \frac{(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^3}+\frac{g \left (a+b \log \left (c x^n\right )\right )^3}{e (d+e x)^2}\right ) \, dx\\ &=\frac{g \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{e}+\frac{(e f-d g) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx}{e}\\ &=-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac{(3 b g n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d e}+\frac{(3 b (e f-d g) n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 e^2}\\ &=-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac{3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d e^2}+\frac{(3 b (e f-d g) n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{2 d e^2}-\frac{(3 b (e f-d g) n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d e}+\frac{\left (6 b^2 g n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac{3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}-\frac{3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d e^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d e^2}+\frac{(3 b (e f-d g) n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 d^2 e^2}-\frac{(3 b (e f-d g) n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{2 d^2 e}+\frac{\left (3 b^2 (e f-d g) n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2 e}+\frac{\left (6 b^3 g n^3\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d e^2}\\ &=-\frac{3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac{3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e^2}-\frac{3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d e^2}-\frac{3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^2 e^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d e^2}+\frac{6 b^3 g n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d e^2}+\frac{(3 (e f-d g)) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 d^2 e^2}+\frac{\left (3 b^2 (e f-d g) n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac{\left (3 b^3 (e f-d g) n^3\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac{3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac{3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e^2}-\frac{3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d e^2}-\frac{3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^2 e^2}+\frac{3 b^3 (e f-d g) n^3 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d e^2}-\frac{3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}+\frac{6 b^3 g n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d e^2}+\frac{\left (3 b^3 (e f-d g) n^3\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^2 e^2}\\ &=-\frac{3 b (e f-d g) n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 e (d+e x)}+\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^2 e^2}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2 (d+e x)^2}+\frac{g x \left (a+b \log \left (c x^n\right )\right )^3}{d e (d+e x)}+\frac{3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^2 e^2}-\frac{3 b g n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d e^2}-\frac{3 b (e f-d g) n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^2 e^2}+\frac{3 b^3 (e f-d g) n^3 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}-\frac{6 b^2 g n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d e^2}-\frac{3 b^2 (e f-d g) n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}+\frac{6 b^3 g n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d e^2}+\frac{3 b^3 (e f-d g) n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^2 e^2}\\ \end{align*}

Mathematica [A]  time = 0.380427, size = 339, normalized size = 1.15 \[ \frac{\frac{(e f-d g) \left (-3 b n (d+e x) \left (\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{e x}{d}+1\right )\right )-2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )\right )-6 b^2 n^2 (d+e x) \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )\right )+(d+e x) \left (a+b \log \left (c x^n\right )\right )^3-3 b n (d+e x) \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+3 b d n \left (a+b \log \left (c x^n\right )\right )^2\right )}{d^2 (d+e x)}+\frac{2 g \left (-6 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^3 n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\left (a+b \log \left (c x^n\right )\right )^2 \left (a+b \log \left (c x^n\right )-3 b n \log \left (\frac{e x}{d}+1\right )\right )\right )}{d}-\frac{(e f-d g) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}-\frac{2 g \left (a+b \log \left (c x^n\right )\right )^3}{d+e x}}{2 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((e*f - d*g)*(a + b*Log[c*x^n])^3)/(d + e*x)^2) - (2*g*(a + b*Log[c*x^n])^3)/(d + e*x) + (2*g*((a + b*Log[c
*x^n])^2*(a + b*Log[c*x^n] - 3*b*n*Log[1 + (e*x)/d]) - 6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 6
*b^3*n^3*PolyLog[3, -((e*x)/d)]))/d + ((e*f - d*g)*(3*b*d*n*(a + b*Log[c*x^n])^2 + (d + e*x)*(a + b*Log[c*x^n]
)^3 - 3*b*n*(d + e*x)*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] - 3*b*n*(d + e*x)*((a + b*Log[c*x^n])*(a + b*Log[c
*x^n] - 2*b*n*Log[1 + (e*x)/d]) - 2*b^2*n^2*PolyLog[2, -((e*x)/d)]) - 6*b^2*n^2*(d + e*x)*((a + b*Log[c*x^n])*
PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)])))/(d^2*(d + e*x)))/(2*e^2)

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Maple [F]  time = 1.284, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{ \left ( ex+d \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \, a^{2} b f n{\left (\frac{1}{d e^{2} x + d^{2} e} - \frac{\log \left (e x + d\right )}{d^{2} e} + \frac{\log \left (x\right )}{d^{2} e}\right )} - \frac{3}{2} \, a^{2} b g n{\left (\frac{1}{e^{3} x + d e^{2}} + \frac{\log \left (e x + d\right )}{d e^{2}} - \frac{\log \left (x\right )}{d e^{2}}\right )} - \frac{3 \,{\left (2 \, e x + d\right )} a^{2} b g \log \left (c x^{n}\right )}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac{{\left (2 \, e x + d\right )} a^{3} g}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac{3 \, a^{2} b f \log \left (c x^{n}\right )}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac{a^{3} f}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac{{\left (2 \, b^{3} e g x +{\left (e f + d g\right )} b^{3}\right )} \log \left (x^{n}\right )^{3}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \int \frac{2 \,{\left (b^{3} e^{2} g \log \left (c\right )^{3} + 3 \, a b^{2} e^{2} g \log \left (c\right )^{2}\right )} x^{2} + 3 \,{\left ({\left (d e f n + d^{2} g n\right )} b^{3} + 2 \,{\left (a b^{2} e^{2} g +{\left (e^{2} g n + e^{2} g \log \left (c\right )\right )} b^{3}\right )} x^{2} +{\left (2 \, a b^{2} e^{2} f +{\left (e^{2} f n + 3 \, d e g n + 2 \, e^{2} f \log \left (c\right )\right )} b^{3}\right )} x\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (b^{3} e^{2} f \log \left (c\right )^{3} + 3 \, a b^{2} e^{2} f \log \left (c\right )^{2}\right )} x + 6 \,{\left ({\left (b^{3} e^{2} g \log \left (c\right )^{2} + 2 \, a b^{2} e^{2} g \log \left (c\right )\right )} x^{2} +{\left (b^{3} e^{2} f \log \left (c\right )^{2} + 2 \, a b^{2} e^{2} f \log \left (c\right )\right )} x\right )} \log \left (x^{n}\right )}{2 \,{\left (e^{5} x^{4} + 3 \, d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + d^{3} e^{2} x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*b*f*n*(1/(d*e^2*x + d^2*e) - log(e*x + d)/(d^2*e) + log(x)/(d^2*e)) - 3/2*a^2*b*g*n*(1/(e^3*x + d*e^2)
+ log(e*x + d)/(d*e^2) - log(x)/(d*e^2)) - 3/2*(2*e*x + d)*a^2*b*g*log(c*x^n)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)
- 1/2*(2*e*x + d)*a^3*g/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) - 3/2*a^2*b*f*log(c*x^n)/(e^3*x^2 + 2*d*e^2*x + d^2*e
) - 1/2*a^3*f/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/2*(2*b^3*e*g*x + (e*f + d*g)*b^3)*log(x^n)^3/(e^4*x^2 + 2*d*e^
3*x + d^2*e^2) + integrate(1/2*(2*(b^3*e^2*g*log(c)^3 + 3*a*b^2*e^2*g*log(c)^2)*x^2 + 3*((d*e*f*n + d^2*g*n)*b
^3 + 2*(a*b^2*e^2*g + (e^2*g*n + e^2*g*log(c))*b^3)*x^2 + (2*a*b^2*e^2*f + (e^2*f*n + 3*d*e*g*n + 2*e^2*f*log(
c))*b^3)*x)*log(x^n)^2 + 2*(b^3*e^2*f*log(c)^3 + 3*a*b^2*e^2*f*log(c)^2)*x + 6*((b^3*e^2*g*log(c)^2 + 2*a*b^2*
e^2*g*log(c))*x^2 + (b^3*e^2*f*log(c)^2 + 2*a*b^2*e^2*f*log(c))*x)*log(x^n))/(e^5*x^4 + 3*d*e^4*x^3 + 3*d^2*e^
3*x^2 + d^3*e^2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{3} g x + a^{3} f +{\left (b^{3} g x + b^{3} f\right )} \log \left (c x^{n}\right )^{3} + 3 \,{\left (a b^{2} g x + a b^{2} f\right )} \log \left (c x^{n}\right )^{2} + 3 \,{\left (a^{2} b g x + a^{2} b f\right )} \log \left (c x^{n}\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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