\(\int \frac {(h+i x)^2 (a+b \log (c (d+e x)^n))^3}{f+g x} \, dx\) [229]

Optimal result

Integrand size = 31, antiderivative size = 660 \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {6 a b^2 i (e h-d i) n^2 x}{e g}+\frac {6 a b^2 i (g h-f i) n^2 x}{g^2}-\frac {6 b^3 i (e h-d i) n^3 x}{e g}-\frac {6 b^3 i (g h-f i) n^3 x}{g^2}-\frac {3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}+\frac {6 b^3 i (e h-d i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac {6 b^3 i (g h-f i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}-\frac {3 b i (e h-d i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^3}+\frac {3 b (g h-f i)^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {6 b^2 (g h-f i)^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {6 b^3 (g h-f i)^2 n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \]

[Out]

6*a*b^2*i*(-d*i+e*h)*n^2*x/e/g+6*a*b^2*i*(-f*i+g*h)*n^2*x/g^2-6*b^3*i*(-d*i+e*h)*n^3*x/e/g-6*b^3*i*(-f*i+g*h)*
n^3*x/g^2-3/8*b^3*i^2*n^3*(e*x+d)^2/e^2/g+6*b^3*i*(-d*i+e*h)*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e^2/g+6*b^3*i*(-f*i+g
*h)*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e/g^2+3/4*b^2*i^2*n^2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2/g-3*b*i*(-d*i+e*h)*n
*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2/g-3*b*i*(-f*i+g*h)*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e/g^2-3/4*b*i^2*n*(e
*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2/g+i*(-d*i+e*h)*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e^2/g+i*(-f*i+g*h)*(e*x+d)*
(a+b*ln(c*(e*x+d)^n))^3/e/g^2+1/2*i^2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^3/e^2/g+(-f*i+g*h)^2*(a+b*ln(c*(e*x+d)^n
))^3*ln(e*(g*x+f)/(-d*g+e*f))/g^3+3*b*(-f*i+g*h)^2*n*(a+b*ln(c*(e*x+d)^n))^2*polylog(2,-g*(e*x+d)/(-d*g+e*f))/
g^3-6*b^2*(-f*i+g*h)^2*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,-g*(e*x+d)/(-d*g+e*f))/g^3+6*b^3*(-f*i+g*h)^2*n^3*p
olylog(4,-g*(e*x+d)/(-d*g+e*f))/g^3

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2465, 2436, 2333, 2332, 2443, 2481, 2421, 2430, 6724, 2448, 2437, 2342, 2341} \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}-\frac {6 b^2 n^2 (g h-f i)^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {6 a b^2 i n^2 x (e h-d i)}{e g}+\frac {6 a b^2 i n^2 x (g h-f i)}{g^2}-\frac {3 b i n (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac {i (d+e x) (e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}-\frac {3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {3 b n (g h-f i)^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^3}+\frac {(g h-f i)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^3}-\frac {3 b i n (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {i (d+e x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {6 b^3 i n^2 (d+e x) (e h-d i) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac {6 b^3 i n^2 (d+e x) (g h-f i) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}+\frac {6 b^3 n^3 (g h-f i)^2 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {6 b^3 i n^3 x (e h-d i)}{e g}-\frac {6 b^3 i n^3 x (g h-f i)}{g^2} \]

[In]

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

[Out]

(6*a*b^2*i*(e*h - d*i)*n^2*x)/(e*g) + (6*a*b^2*i*(g*h - f*i)*n^2*x)/g^2 - (6*b^3*i*(e*h - d*i)*n^3*x)/(e*g) -
(6*b^3*i*(g*h - f*i)*n^3*x)/g^2 - (3*b^3*i^2*n^3*(d + e*x)^2)/(8*e^2*g) + (6*b^3*i*(e*h - d*i)*n^2*(d + e*x)*L
og[c*(d + e*x)^n])/(e^2*g) + (6*b^3*i*(g*h - f*i)*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (3*b^2*i^2*n^2*(
d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2*g) - (3*b*i*(e*h - d*i)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2
)/(e^2*g) - (3*b*i*(g*h - f*i)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) - (3*b*i^2*n*(d + e*x)^2*(a +
b*Log[c*(d + e*x)^n])^2)/(4*e^2*g) + (i*(e*h - d*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e^2*g) + (i*(g*h
- f*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e*g^2) + (i^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^
2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (3*b*(g*h - f*i)^2*n*
(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3 - (6*b^2*(g*h - f*i)^2*n^2*(a + b*L
og[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^3 + (6*b^3*(g*h - f*i)^2*n^3*PolyLog[4, -((g*(d
+ e*x))/(e*f - d*g))])/g^3

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 2436

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

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 2448

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

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*} \text {integral}& = \int \left (\frac {i (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2 (f+g x)}+\frac {i (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}\right ) \, dx \\ & = \frac {i \int (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g}+\frac {(i (g h-f i)) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{g^2}+\frac {(g h-f i)^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx}{g^2} \\ & = \frac {(g h-f i)^2 \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^3}+\frac {i \int \left (\frac {(e h-d i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx}{g}+\frac {(i (g h-f i)) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e g^2}-\frac {\left (3 b e (g h-f i)^2 n\right ) \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}{g^3} \\ & = \frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {(g h-f i)^2 \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^3}+\frac {i^2 \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e g}+\frac {(i (e h-d i)) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e g}-\frac {(3 b i (g h-f i) n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e g^2}-\frac {\left (3 b (g h-f i)^2 n\right ) \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 )}{g^3} \\ & = -\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {(g h-f i)^2 \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^3}+\frac {3 b (g h-f i)^2 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^3}+\frac {i^2 \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^2 g}+\frac {(i (e h-d i)) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^2 g}+\frac {\left (6 b^2 i (g h-f i) n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e g^2}-\frac {\left (6 b^2 (g h-f i)^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 )}{g^3} \\ & = \frac {6 a b^2 i (g h-f i) n^2 x}{g^2}-\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {(g h-f i)^2 \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^3}+\frac {3 b (g h-f i)^2 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^3}-\frac {6 b^2 (g h-f i)^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^3}-\frac {\left (3 b i^2 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2 g}-\frac {(3 b i (e h-d i) n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g}+\frac {\left (6 b^3 i (g h-f i) n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}+\frac {\left (6 b^3 (g h-f i)^2 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 )}{g^3} \\ & = \frac {6 a b^2 i (g h-f i) n^2 x}{g^2}-\frac {6 b^3 i (g h-f i) n^3 x}{g^2}+\frac {6 b^3 i (g h-f i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {3 b i (e h-d i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {(g h-f i)^2 \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^3}+\frac {3 b (g h-f i)^2 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^3}-\frac {6 b^2 (g h-f i)^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^3}+\frac {6 b^3 (g h-f i)^2 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {\left (3 b^2 i^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2 g}+\frac {\left (6 b^2 i (e h-d i) n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g} \\ & = \frac {6 a b^2 i (e h-d i) n^2 x}{e g}+\frac {6 a b^2 i (g h-f i) n^2 x}{g^2}-\frac {6 b^3 i (g h-f i) n^3 x}{g^2}-\frac {3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}+\frac {6 b^3 i (g h-f i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}-\frac {3 b i (e h-d i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {(g h-f i)^2 \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^3}+\frac {3 b (g h-f i)^2 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^3}-\frac {6 b^2 (g h-f i)^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^3}+\frac {6 b^3 (g h-f i)^2 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}+\frac {\left (6 b^3 i (e h-d i) n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g} \\ & = \frac {6 a b^2 i (e h-d i) n^2 x}{e g}+\frac {6 a b^2 i (g h-f i) n^2 x}{g^2}-\frac {6 b^3 i (e h-d i) n^3 x}{e g}-\frac {6 b^3 i (g h-f i) n^3 x}{g^2}-\frac {3 b^3 i^2 n^3 (d+e x)^2}{8 e^2 g}+\frac {6 b^3 i (e h-d i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac {6 b^3 i (g h-f i) n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {3 b^2 i^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 g}-\frac {3 b i (e h-d i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac {3 b i (g h-f i) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g^2}-\frac {3 b i^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 g}+\frac {i (e h-d i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 g}+\frac {i (g h-f i) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g^2}+\frac {i^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 g}+\frac {(g h-f i)^2 \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^3}+\frac {3 b (g h-f i)^2 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^3}-\frac {6 b^2 (g h-f i)^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^3}+\frac {6 b^3 (g h-f i)^2 n^3 \text {Li}_4\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1521\) vs. \(2(660)=1320\).

Time = 0.51 (sec) , antiderivative size = 1521, normalized size of antiderivative = 2.30 \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {8 e^2 g i (2 g h-f i) x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+4 e^2 g^2 i^2 x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+8 e^2 (g h-f i)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)+24 b e^2 g^2 h^2 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) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+6 b i^2 n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (e g (e x (4 f-g x)+2 d (2 f+g x))-2 \log (d+e x) \left (g (d+e x) (2 e f+d g-e g x)-2 e^2 f^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+4 e^2 f^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )-48 b e g h i n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (-g (d+e x) (-1+\log (d+e x))+e f \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )+48 b^2 e g h i n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (g \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )-e f \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )-6 b^2 i^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (4 e f g \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )+g^2 \left (e x (6 d-e x)+\left (-6 d^2-4 d e x+2 e^2 x^2\right ) \log (d+e x)+2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)\right )-4 e^2 f^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )+48 b^2 e^2 g^2 h^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 )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )+8 b^3 e^2 g^2 h^2 n^3 \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )-16 b^3 e g h i n^3 \left (g \left (6 e x-6 (d+e x) \log (d+e x)+3 (d+e x) \log ^2(d+e x)-(d+e x) \log ^3(d+e x)\right )+e f \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )+b^3 i^2 n^3 \left (8 e f g \left (6 e x-6 (d+e x) \log (d+e x)+3 (d+e x) \log ^2(d+e x)-(d+e x) \log ^3(d+e x)\right )-g^2 \left (3 e x (-14 d+e x)+6 \left (7 d^2+6 d e x-e^2 x^2\right ) \log (d+e x)-6 \left (3 d^2+2 d e x-e^2 x^2\right ) \log ^2(d+e x)+4 \left (d^2-e^2 x^2\right ) \log ^3(d+e x)\right )+8 e^2 f^2 \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )}{8 e^2 g^3} \]

[In]

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

[Out]

(8*e^2*g*i*(2*g*h - f*i)*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 + 4*e^2*g^2*i^2*x^2*(a - b*n*Log[d
+ e*x] + b*Log[c*(d + e*x)^n])^3 + 8*e^2*(g*h - f*i)^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*Log[f +
g*x] + 24*b*e^2*g^2*h^2*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)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 6*b*i^2*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]
)^2*(e*g*(e*x*(4*f - g*x) + 2*d*(2*f + g*x)) - 2*Log[d + e*x]*(g*(d + e*x)*(2*e*f + d*g - e*g*x) - 2*e^2*f^2*L
og[(e*(f + g*x))/(e*f - d*g)]) + 4*e^2*f^2*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) - 48*b*e*g*h*i*n*(a - b*n
*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(-(g*(d + e*x)*(-1 + Log[d + e*x])) + e*f*(Log[d + e*x]*Log[(e*(f + g*
x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)])) + 48*b^2*e*g*h*i*n^2*(a - b*n*Log[d + e*x] + b*L
og[c*(d + e*x)^n])*(g*(2*e*x - 2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) - e*f*(Log[d + e*x]^2*Log[
(e*(f + g*x))/(e*f - d*g)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*
x))/(-(e*f) + d*g)])) - 6*b^2*i^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(4*e*f*g*(2*e*x - 2*(d + e
*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) + g^2*(e*x*(6*d - e*x) + (-6*d^2 - 4*d*e*x + 2*e^2*x^2)*Log[d + e
*x] + 2*(d^2 - e^2*x^2)*Log[d + e*x]^2) - 4*e^2*f^2*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*Log[d +
e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)])) + 48*b^2*e^2*g^2
*h^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 + Lo
g[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]) + 8*b^3*e^2*g^
2*h^2*n^3*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) +
d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)])
- 16*b^3*e*g*h*i*n^3*(g*(6*e*x - 6*(d + e*x)*Log[d + e*x] + 3*(d + e*x)*Log[d + e*x]^2 - (d + e*x)*Log[d + e*
x]^3) + e*f*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f)
+ d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)
])) + b^3*i^2*n^3*(8*e*f*g*(6*e*x - 6*(d + e*x)*Log[d + e*x] + 3*(d + e*x)*Log[d + e*x]^2 - (d + e*x)*Log[d +
e*x]^3) - g^2*(3*e*x*(-14*d + e*x) + 6*(7*d^2 + 6*d*e*x - e^2*x^2)*Log[d + e*x] - 6*(3*d^2 + 2*d*e*x - e^2*x^2
)*Log[d + e*x]^2 + 4*(d^2 - e^2*x^2)*Log[d + e*x]^3) + 8*e^2*f^2*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)
] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f
) + d*g)] + 6*PolyLog[4, (g*(d + e*x))/(-(e*f) + d*g)])))/(8*e^2*g^3)

Maple [F]

\[\int \frac {\left (i x +h \right )^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}{g x +f}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]

[In]

integrate((i*x+h)^2*(a+b*log(c*(e*x+d)^n))^3/(g*x+f),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (h + i x\right )^{2}}{f + g x}\, dx \]

[In]

integrate((i*x+h)**2*(a+b*ln(c*(e*x+d)**n))**3/(g*x+f),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))**3*(h + i*x)**2/(f + g*x), x)

Maxima [F]

\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]

[In]

integrate((i*x+h)^2*(a+b*log(c*(e*x+d)^n))^3/(g*x+f),x, algorithm="maxima")

[Out]

2*a^3*h*i*(x/g - f*log(g*x + f)/g^2) + 1/2*a^3*i^2*(2*f^2*log(g*x + f)/g^3 + (g*x^2 - 2*f*x)/g^2) + a^3*h^2*lo
g(g*x + f)/g + integrate((b^3*h^2*log(c)^3 + 3*a*b^2*h^2*log(c)^2 + 3*a^2*b*h^2*log(c) + (b^3*i^2*x^2 + 2*b^3*
h*i*x + b^3*h^2)*log((e*x + d)^n)^3 + (b^3*i^2*log(c)^3 + 3*a*b^2*i^2*log(c)^2 + 3*a^2*b*i^2*log(c))*x^2 + 3*(
b^3*h^2*log(c) + a*b^2*h^2 + (b^3*i^2*log(c) + a*b^2*i^2)*x^2 + 2*(b^3*h*i*log(c) + a*b^2*h*i)*x)*log((e*x + d
)^n)^2 + 2*(b^3*h*i*log(c)^3 + 3*a*b^2*h*i*log(c)^2 + 3*a^2*b*h*i*log(c))*x + 3*(b^3*h^2*log(c)^2 + 2*a*b^2*h^
2*log(c) + a^2*b*h^2 + (b^3*i^2*log(c)^2 + 2*a*b^2*i^2*log(c) + a^2*b*i^2)*x^2 + 2*(b^3*h*i*log(c)^2 + 2*a*b^2
*h*i*log(c) + a^2*b*h*i)*x)*log((e*x + d)^n))/(g*x + f), x)

Giac [F]

\[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]

[In]

integrate((i*x+h)^2*(a+b*log(c*(e*x+d)^n))^3/(g*x+f),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int \frac {{\left (h+i\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{f+g\,x} \,d x \]

[In]

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

[Out]

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