3.190 \(\int \frac{(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx\)

Optimal. Leaf size=485 \[ \frac{2 b f^2 \text{PolyLog}\left (2,-\frac{f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{b^2 f^2 \text{PolyLog}\left (2,-\frac{f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac{2 b^2 f^2 \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac{2 b f^2 \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{f^2 \log \left (\frac{f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^3}+\frac{b f^2 \log \left (\frac{f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{f i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^3}+\frac{b f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac{(a+b \log (c (e+f x)))^2}{2 d (h+i x)^2 (f h-e i)}-\frac{b^2 f^2 \log (h+i x)}{d (f h-e i)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.09201, antiderivative size = 453, normalized size of antiderivative = 0.93, number of steps used = 21, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {2411, 12, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ -\frac{2 b f^2 \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}+\frac{3 b^2 f^2 \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac{f^2 (a+b \log (c (e+f x)))^3}{3 b d (f h-e i)^3}-\frac{f^2 \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^3}-\frac{f^2 (a+b \log (c (e+f x)))^2}{2 d (f h-e i)^3}+\frac{3 b f^2 \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac{f i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^3}+\frac{(a+b \log (c (e+f x)))^2}{2 d (h+i x)^2 (f h-e i)}+\frac{b f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}-\frac{b^2 f^2 \log (h+i x)}{d (f h-e i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(a+b \log (c (e+f x)))^2}{(h+190 x)^3 (d e+d f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{d x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^3} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (190 e-f h)}+\frac{190 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^3} \, dx,x,e+f x\right )}{d f (190 e-f h)}\\ &=-\frac{(a+b \log (c (e+f x)))^2}{2 d (190 e-f h) (h+190 x)^2}-\frac{190 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (190 e-f h)^2}+\frac{f \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )} \, dx,x,e+f x\right )}{d (190 e-f h)^2}+\frac{b \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (190 e-f h)}\\ &=-\frac{(a+b \log (c (e+f x)))^2}{2 d (190 e-f h) (h+190 x)^2}+\frac{190 f (e+f x) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3 (h+190 x)}+\frac{(190 f) \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\frac{-190 e+f h}{f}+\frac{190 x}{f}} \, dx,x,e+f x\right )}{d (190 e-f h)^3}-\frac{(380 b f) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\frac{-190 e+f h}{f}+\frac{190 x}{f}} \, dx,x,e+f x\right )}{d (190 e-f h)^3}-\frac{f^2 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}+\frac{(190 b) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )^2} \, dx,x,e+f x\right )}{d (190 e-f h)^2}-\frac{(b f) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x \left (\frac{-190 e+f h}{f}+\frac{190 x}{f}\right )} \, dx,x,e+f x\right )}{d (190 e-f h)^2}\\ &=-\frac{190 b f (e+f x) (a+b \log (c (e+f x)))}{d (190 e-f h)^3 (h+190 x)}-\frac{2 b f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))}{d (190 e-f h)^3}-\frac{(a+b \log (c (e+f x)))^2}{2 d (190 e-f h) (h+190 x)^2}+\frac{190 f (e+f x) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3 (h+190 x)}+\frac{f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3}-\frac{(190 b f) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\frac{-190 e+f h}{f}+\frac{190 x}{f}} \, dx,x,e+f x\right )}{d (190 e-f h)^3}+\frac{\left (190 b^2 f\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{-190 e+f h}{f}+\frac{190 x}{f}} \, dx,x,e+f x\right )}{d (190 e-f h)^3}-\frac{f^2 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d (190 e-f h)^3}+\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}-\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{(a+b \log (c x)) \log \left (1+\frac{190 x}{-190 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}+\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{190 x}{-190 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}\\ &=\frac{b^2 f^2 \log (h+190 x)}{d (190 e-f h)^3}-\frac{190 b f (e+f x) (a+b \log (c (e+f x)))}{d (190 e-f h)^3 (h+190 x)}-\frac{3 b f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))}{d (190 e-f h)^3}+\frac{f^2 (a+b \log (c (e+f x)))^2}{2 d (190 e-f h)^3}-\frac{(a+b \log (c (e+f x)))^2}{2 d (190 e-f h) (h+190 x)^2}+\frac{190 f (e+f x) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3 (h+190 x)}+\frac{f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3}-\frac{f^2 (a+b \log (c (e+f x)))^3}{3 b d (190 e-f h)^3}-\frac{2 b^2 f^2 \text{Li}_2\left (\frac{190 (e+f x)}{190 e-f h}\right )}{d (190 e-f h)^3}+\frac{2 b f^2 (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{190 (e+f x)}{190 e-f h}\right )}{d (190 e-f h)^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{190 x}{-190 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}-\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{190 x}{-190 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (190 e-f h)^3}\\ &=\frac{b^2 f^2 \log (h+190 x)}{d (190 e-f h)^3}-\frac{190 b f (e+f x) (a+b \log (c (e+f x)))}{d (190 e-f h)^3 (h+190 x)}-\frac{3 b f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))}{d (190 e-f h)^3}+\frac{f^2 (a+b \log (c (e+f x)))^2}{2 d (190 e-f h)^3}-\frac{(a+b \log (c (e+f x)))^2}{2 d (190 e-f h) (h+190 x)^2}+\frac{190 f (e+f x) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3 (h+190 x)}+\frac{f^2 \log \left (-\frac{f (h+190 x)}{190 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (190 e-f h)^3}-\frac{f^2 (a+b \log (c (e+f x)))^3}{3 b d (190 e-f h)^3}-\frac{3 b^2 f^2 \text{Li}_2\left (\frac{190 (e+f x)}{190 e-f h}\right )}{d (190 e-f h)^3}+\frac{2 b f^2 (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{190 (e+f x)}{190 e-f h}\right )}{d (190 e-f h)^3}-\frac{2 b^2 f^2 \text{Li}_3\left (\frac{190 (e+f x)}{190 e-f h}\right )}{d (190 e-f h)^3}\\ \end{align*}

Mathematica [A]  time = 0.885183, size = 680, normalized size = 1.4 \[ \frac{6 a b \left (-2 f^2 (h+i x)^2 \left (\text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )+\log (c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )\right )+f^2 (h+i x)^2 \log ^2(c (e+f x))+(f h-e i)^2 \log (c (e+f x))-2 f (h+i x) ((e i-f h) \log (c (e+f x))+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))-f (h+i x) (f (h+i x) \log (e+f x)-e i-f (h+i x) \log (h+i x)+f h)\right )+b^2 \left (-6 f^2 (h+i x)^2 \left (2 \log (c (e+f x)) \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )-2 \text{PolyLog}\left (3,\frac{i (e+f x)}{e i-f h}\right )+\log ^2(c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )\right )-6 f (h+i x) \left (\log (c (e+f x)) \left (i (e+f x) \log (c (e+f x))-2 f (h+i x) \log \left (\frac{f (h+i x)}{f h-e i}\right )\right )-2 f (h+i x) \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )\right )+6 f^2 (h+i x)^2 \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )+2 f^2 (h+i x)^2 \log ^3(c (e+f x))-3 f^2 (h+i x)^2 \log ^2(c (e+f x))+6 f^2 (h+i x)^2 \log (c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )+3 (f h-e i)^2 \log ^2(c (e+f x))-6 f (h+i x) (f h-e i) \log (c (e+f x))+6 f^2 (h+i x)^2 \log (e+f x)-6 f^2 (h+i x)^2 \log (h+i x)\right )+6 a^2 f^2 (h+i x)^2 \log (e+f x)+6 a^2 f (h+i x) (f h-e i)+3 a^2 (f h-e i)^2-6 a^2 f^2 (h+i x)^2 \log (h+i x)}{6 d (h+i x)^2 (f h-e i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 2.142, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) ^{2}}{ \left ( dfx+de \right ) \left ( ix+h \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 2.04545, size = 1716, normalized size = 3.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*f^2*log(f*x + e)/(d*f^3*h^3 - 3*d*e*f^2*h^2*i + 3*d*e^2*f*h*i^2 - d*e^3*i^3) - 2*f^2*log(i*x + h)/(d*f^
3*h^3 - 3*d*e*f^2*h^2*i + 3*d*e^2*f*h*i^2 - d*e^3*i^3) + (2*f*i*x + 3*f*h - e*i)/(d*f^2*h^4 - 2*d*e*f*h^3*i +
d*e^2*h^2*i^2 + (d*f^2*h^2*i^2 - 2*d*e*f*h*i^3 + d*e^2*i^4)*x^2 + 2*(d*f^2*h^3*i - 2*d*e*f*h^2*i^2 + d*e^2*h*i
^3)*x))*a^2 - (log(f*x + e)^2*log((f*i*x + e*i)/(f*h - e*i) + 1) + 2*dilog(-(f*i*x + e*i)/(f*h - e*i))*log(f*x
+ e) - 2*polylog(3, -(f*i*x + e*i)/(f*h - e*i)))*b^2*f^2/((f^3*h^3 - 3*e*f^2*h^2*i + 3*e^2*f*h*i^2 - e^3*i^3)
*d) + 1/6*(2*(b^2*f^2*i^2*x^2 + 2*b^2*f^2*h*i*x + b^2*f^2*h^2)*log(f*x + e)^3 - 6*(f^2*h^2 - e*f*h*i - (3*f^2*
h^2 - 4*e*f*h*i + e^2*i^2)*log(c))*a*b + 3*((3*f^2*h^2 - 4*e*f*h*i + e^2*i^2)*log(c)^2 - 2*(f^2*h^2 - e*f*h*i)
*log(c))*b^2 + 3*(2*a*b*f^2*h^2 + (2*f^2*h^2*log(c) - 4*e*f*h*i + e^2*i^2)*b^2 + (2*a*b*f^2*i^2 + (2*f^2*i^2*l
og(c) - 3*f^2*i^2)*b^2)*x^2 + 2*(2*a*b*f^2*h*i + (2*f^2*h*i*log(c) - 2*f^2*h*i - e*f*i^2)*b^2)*x)*log(f*x + e)
^2 - 6*((f^2*h*i - e*f*i^2 - 2*(f^2*h*i - e*f*i^2)*log(c))*a*b - ((f^2*h*i - e*f*i^2)*log(c)^2 - (f^2*h*i - e*
f*i^2)*log(c))*b^2)*x + 6*((2*f^2*h^2*log(c) - 4*e*f*h*i + e^2*i^2)*a*b + (f^2*h^2*log(c)^2 + e*f*h*i - (4*e*f
*h*i - e^2*i^2)*log(c))*b^2 + ((2*f^2*i^2*log(c) - 3*f^2*i^2)*a*b + (f^2*i^2*log(c)^2 - 3*f^2*i^2*log(c) + f^2
*i^2)*b^2)*x^2 + (2*(2*f^2*h*i*log(c) - 2*f^2*h*i - e*f*i^2)*a*b + (2*f^2*h*i*log(c)^2 + f^2*h*i + e*f*i^2 - 2
*(2*f^2*h*i + e*f*i^2)*log(c))*b^2)*x)*log(f*x + e))/((f^3*h^3*i^2 - 3*e*f^2*h^2*i^3 + 3*e^2*f*h*i^4 - e^3*i^5
)*d*x^2 + 2*(f^3*h^4*i - 3*e*f^2*h^3*i^2 + 3*e^2*f*h^2*i^3 - e^3*h*i^4)*d*x + (f^3*h^5 - 3*e*f^2*h^4*i + 3*e^2
*f*h^3*i^2 - e^3*h^2*i^3)*d) - (2*a*b*f^2 + (2*f^2*log(c) - 3*f^2)*b^2)*(log(f*x + e)*log((f*i*x + e*i)/(f*h -
e*i) + 1) + dilog(-(f*i*x + e*i)/(f*h - e*i)))/((f^3*h^3 - 3*e*f^2*h^2*i + 3*e^2*f*h*i^2 - e^3*i^3)*d) - ((2*
f^2*log(c) - 3*f^2)*a*b + (f^2*log(c)^2 - 3*f^2*log(c) + f^2)*b^2)*log(i*x + h)/((f^3*h^3 - 3*e*f^2*h^2*i + 3*
e^2*f*h*i^2 - e^3*i^3)*d)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c f x + c e\right )^{2} + 2 \, a b \log \left (c f x + c e\right ) + a^{2}}{d f i^{3} x^{4} + d e h^{3} +{\left (3 \, d f h i^{2} + d e i^{3}\right )} x^{3} + 3 \,{\left (d f h^{2} i + d e h i^{2}\right )} x^{2} +{\left (d f h^{3} + 3 \, d e h^{2} i\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*f*x + c*e)^2 + 2*a*b*log(c*f*x + c*e) + a^2)/(d*f*i^3*x^4 + d*e*h^3 + (3*d*f*h*i^2 + d*e*i
^3)*x^3 + 3*(d*f*h^2*i + d*e*h*i^2)*x^2 + (d*f*h^3 + 3*d*e*h^2*i)*x), x)

________________________________________________________________________________________

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*(f*x+e)))**2/(d*f*x+d*e)/(i*x+h)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )}{\left (i x + h\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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