3.2.90 \(\int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx\) [190]

Optimal. Leaf size=485 \[ \frac {b f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {(a+b \log (c (e+f x)))^2}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^3 (h+i x)}-\frac {b^2 f^2 \log (h+i x)}{d (f h-e i)^3}+\frac {2 b f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac {b f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {b^2 f^2 \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {2 b f^2 (a+b \log (c (e+f x))) \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \text {Li}_2\left (-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]

[Out]

b*f*i*(f*x+e)*(a+b*ln(c*(f*x+e)))/d/(-e*i+f*h)^3/(i*x+h)+1/2*(a+b*ln(c*(f*x+e)))^2/d/(-e*i+f*h)/(i*x+h)^2-f*i*
(f*x+e)*(a+b*ln(c*(f*x+e)))^2/d/(-e*i+f*h)^3/(i*x+h)-b^2*f^2*ln(i*x+h)/d/(-e*i+f*h)^3+2*b*f^2*(a+b*ln(c*(f*x+e
)))*ln(f*(i*x+h)/(-e*i+f*h))/d/(-e*i+f*h)^3+b*f^2*(a+b*ln(c*(f*x+e)))*ln(1+(-e*i+f*h)/i/(f*x+e))/d/(-e*i+f*h)^
3-f^2*(a+b*ln(c*(f*x+e)))^2*ln(1+(-e*i+f*h)/i/(f*x+e))/d/(-e*i+f*h)^3-b^2*f^2*polylog(2,(e*i-f*h)/i/(f*x+e))/d
/(-e*i+f*h)^3+2*b*f^2*(a+b*ln(c*(f*x+e)))*polylog(2,(e*i-f*h)/i/(f*x+e))/d/(-e*i+f*h)^3+2*b^2*f^2*polylog(2,-i
*(f*x+e)/(-e*i+f*h))/d/(-e*i+f*h)^3+2*b^2*f^2*polylog(3,(e*i-f*h)/i/(f*x+e))/d/(-e*i+f*h)^3

________________________________________________________________________________________

Rubi [A]
time = 0.74, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2458, 12, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \begin {gather*} \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} \end {gather*}

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)) + (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)

Rule 12

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

Rule 31

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

Rule 2351

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 2354

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 2355

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 2356

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 2379

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

Rule 2389

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 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 2438

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 2458

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 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 {(a+b \log (c (e+f x)))^2}{(h+190 x)^3 (d e+d f x)} \, dx &=\frac {\text {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 {\text {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 {\text {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 \text {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 \text {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 \text {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 \text {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) \text {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) \text {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 \text {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) \text {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) \text {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) \text {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 ) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.61, size = 680, normalized size = 1.40 \begin {gather*} \frac {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 \left ((f h-e i)^2 \log (c (e+f x))+f^2 (h+i x)^2 \log ^2(c (e+f x))-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 \left (\log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+\text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )\right )\right )+b^2 \left (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 ^2(c (e+f x))-3 f^2 (h+i x)^2 \log ^2(c (e+f x))+2 f^2 (h+i x)^2 \log ^3(c (e+f x))-6 f^2 (h+i x)^2 \log (h+i 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 )+6 f^2 (h+i x)^2 \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\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 {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )\right )-6 f^2 (h+i x)^2 \left (\log ^2(c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+2 \log (c (e+f x)) \text {Li}_2\left (\frac {i (e+f x)}{-f h+e i}\right )-2 \text {Li}_3\left (\frac {i (e+f x)}{-f h+e i}\right )\right )\right )}{6 d (f h-e i)^3 (h+i x)^2} \end {gather*}

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 = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{2}}{\left (d f x +e d \right ) \left (i x +h \right )^{3}}\, dx\]

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (476) = 952\).
time = 2.61, size = 1068, normalized size = 2.20 \begin {gather*} -\frac {{\left (\log \left (f x + e\right )^{2} \log \left (\frac {i \, f x + i \, e}{f h - i \, e} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {i \, f x + i \, e}{f h - i \, e}\right ) \log \left (f x + e\right ) - 2 \, {\rm Li}_{3}(-\frac {i \, f x + i \, e}{f h - i \, e})\right )} b^{2} f^{2}}{d f^{3} h^{3} - 3 i \, d f^{2} h^{2} e - 3 \, d f h e^{2} + i \, d e^{3}} - \frac {{\left (2 \, a b f^{2} + {\left (2 \, f^{2} \log \left (c\right ) - 3 \, f^{2}\right )} b^{2}\right )} {\left (\log \left (f x + e\right ) \log \left (\frac {i \, f x + i \, e}{f h - i \, e} + 1\right ) + {\rm Li}_2\left (-\frac {i \, f x + i \, e}{f h - i \, e}\right )\right )}}{d f^{3} h^{3} - 3 i \, d f^{2} h^{2} e - 3 \, d f h e^{2} + i \, d e^{3}} - \frac {{\left (a^{2} f^{2} + {\left (2 \, f^{2} \log \left (c\right ) - 3 \, f^{2}\right )} a b + {\left (f^{2} \log \left (c\right )^{2} - 3 \, f^{2} \log \left (c\right ) + f^{2}\right )} b^{2}\right )} \log \left (h + i \, x\right )}{d f^{3} h^{3} - 3 i \, d f^{2} h^{2} e - 3 \, d f h e^{2} + i \, d e^{3}} + \frac {8 \, {\left (9 i \, a^{2} f^{2} h^{2} + 2 \, {\left (i \, b^{2} f^{2} h^{2} - 2 \, b^{2} f^{2} h x - i \, b^{2} f^{2} x^{2}\right )} \log \left (f x + e\right )^{3} + 6 \, {\left (3 i \, f^{2} h^{2} \log \left (c\right ) - i \, f^{2} h^{2}\right )} a b + 3 \, {\left (3 i \, f^{2} h^{2} \log \left (c\right )^{2} - 2 i \, f^{2} h^{2} \log \left (c\right )\right )} b^{2} + 3 \, {\left (2 i \, b^{2} f^{2} h^{2} \log \left (c\right ) + 2 i \, a b f^{2} h^{2} + 4 \, b^{2} f h e + {\left (-2 i \, a b f^{2} + {\left (-2 i \, f^{2} \log \left (c\right ) + 3 i \, f^{2}\right )} b^{2}\right )} x^{2} - i \, b^{2} e^{2} - 2 \, {\left (2 \, a b f^{2} h - i \, b^{2} f e + 2 \, {\left (f^{2} h \log \left (c\right ) - f^{2} h\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )^{2} - 6 \, {\left (a^{2} f^{2} h + {\left (2 \, f^{2} h \log \left (c\right ) - f^{2} h\right )} a b + {\left (f^{2} h \log \left (c\right )^{2} - f^{2} h \log \left (c\right )\right )} b^{2} - {\left ({\left (2 i \, f \log \left (c\right ) - i \, f\right )} a b + {\left (i \, f \log \left (c\right )^{2} - i \, f \log \left (c\right )\right )} b^{2} + i \, a^{2} f\right )} e\right )} x + 3 \, {\left (-i \, b^{2} \log \left (c\right )^{2} - 2 i \, a b \log \left (c\right ) - i \, a^{2}\right )} e^{2} + 6 \, {\left (2 \, a^{2} f h + {\left (4 \, f h \log \left (c\right ) - f h\right )} a b + {\left (2 \, f h \log \left (c\right )^{2} - f h \log \left (c\right )\right )} b^{2}\right )} e + 6 \, {\left (i \, b^{2} f^{2} h^{2} \log \left (c\right )^{2} + 2 i \, a b f^{2} h^{2} \log \left (c\right ) + i \, a^{2} f^{2} h^{2} + {\left (-i \, a^{2} f^{2} + {\left (-2 i \, f^{2} \log \left (c\right ) + 3 i \, f^{2}\right )} a b + {\left (-i \, f^{2} \log \left (c\right )^{2} + 3 i \, f^{2} \log \left (c\right ) - i \, f^{2}\right )} b^{2}\right )} x^{2} - {\left (2 \, a^{2} f^{2} h + 4 \, {\left (f^{2} h \log \left (c\right ) - f^{2} h\right )} a b + {\left (2 \, f^{2} h \log \left (c\right )^{2} - 4 \, f^{2} h \log \left (c\right ) + f^{2} h\right )} b^{2} - {\left ({\left (2 i \, f \log \left (c\right ) - i \, f\right )} b^{2} + 2 i \, a b f\right )} e\right )} x + {\left (-i \, b^{2} \log \left (c\right ) - i \, a b\right )} e^{2} + {\left (4 \, a b f h + {\left (4 \, f h \log \left (c\right ) - f h\right )} b^{2}\right )} e\right )} \log \left (f x + e\right )\right )}}{48 i \, d f^{3} h^{5} + 144 \, d f^{2} h^{4} e - 144 i \, d f h^{3} e^{2} - 48 \, d h^{2} e^{3} - 48 \, {\left (i \, d f^{3} h^{3} + 3 \, d f^{2} h^{2} e - 3 i \, d f h e^{2} - d e^{3}\right )} x^{2} - 96 \, {\left (d f^{3} h^{4} - 3 i \, d f^{2} h^{3} e - 3 \, d f h^{2} e^{2} + i \, d h e^{3}\right )} x} \end {gather*}

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]

-(log(f*x + e)^2*log((I*f*x + I*e)/(f*h - I*e) + 1) + 2*dilog(-(I*f*x + I*e)/(f*h - I*e))*log(f*x + e) - 2*pol
ylog(3, -(I*f*x + I*e)/(f*h - I*e)))*b^2*f^2/(d*f^3*h^3 - 3*I*d*f^2*h^2*e - 3*d*f*h*e^2 + I*d*e^3) - (2*a*b*f^
2 + (2*f^2*log(c) - 3*f^2)*b^2)*(log(f*x + e)*log((I*f*x + I*e)/(f*h - I*e) + 1) + dilog(-(I*f*x + I*e)/(f*h -
I*e)))/(d*f^3*h^3 - 3*I*d*f^2*h^2*e - 3*d*f*h*e^2 + I*d*e^3) - (a^2*f^2 + (2*f^2*log(c) - 3*f^2)*a*b + (f^2*l
og(c)^2 - 3*f^2*log(c) + f^2)*b^2)*log(h + I*x)/(d*f^3*h^3 - 3*I*d*f^2*h^2*e - 3*d*f*h*e^2 + I*d*e^3) + 8*(9*I
*a^2*f^2*h^2 + 2*(I*b^2*f^2*h^2 - 2*b^2*f^2*h*x - I*b^2*f^2*x^2)*log(f*x + e)^3 + 6*(3*I*f^2*h^2*log(c) - I*f^
2*h^2)*a*b + 3*(3*I*f^2*h^2*log(c)^2 - 2*I*f^2*h^2*log(c))*b^2 + 3*(2*I*b^2*f^2*h^2*log(c) + 2*I*a*b*f^2*h^2 +
4*b^2*f*h*e + (-2*I*a*b*f^2 + (-2*I*f^2*log(c) + 3*I*f^2)*b^2)*x^2 - I*b^2*e^2 - 2*(2*a*b*f^2*h - I*b^2*f*e +
2*(f^2*h*log(c) - f^2*h)*b^2)*x)*log(f*x + e)^2 - 6*(a^2*f^2*h + (2*f^2*h*log(c) - f^2*h)*a*b + (f^2*h*log(c)
^2 - f^2*h*log(c))*b^2 - ((2*I*f*log(c) - I*f)*a*b + (I*f*log(c)^2 - I*f*log(c))*b^2 + I*a^2*f)*e)*x + 3*(-I*b
^2*log(c)^2 - 2*I*a*b*log(c) - I*a^2)*e^2 + 6*(2*a^2*f*h + (4*f*h*log(c) - f*h)*a*b + (2*f*h*log(c)^2 - f*h*lo
g(c))*b^2)*e + 6*(I*b^2*f^2*h^2*log(c)^2 + 2*I*a*b*f^2*h^2*log(c) + I*a^2*f^2*h^2 + (-I*a^2*f^2 + (-2*I*f^2*lo
g(c) + 3*I*f^2)*a*b + (-I*f^2*log(c)^2 + 3*I*f^2*log(c) - I*f^2)*b^2)*x^2 - (2*a^2*f^2*h + 4*(f^2*h*log(c) - f
^2*h)*a*b + (2*f^2*h*log(c)^2 - 4*f^2*h*log(c) + f^2*h)*b^2 - ((2*I*f*log(c) - I*f)*b^2 + 2*I*a*b*f)*e)*x + (-
I*b^2*log(c) - I*a*b)*e^2 + (4*a*b*f*h + (4*f*h*log(c) - f*h)*b^2)*e)*log(f*x + e))/(48*I*d*f^3*h^5 + 144*d*f^
2*h^4*e - 144*I*d*f*h^3*e^2 - 48*d*h^2*e^3 - 48*(I*d*f^3*h^3 + 3*d*f^2*h^2*e - 3*I*d*f*h*e^2 - d*e^3)*x^2 - 96
*(d*f^3*h^4 - 3*I*d*f^2*h^3*e - 3*d*f*h^2*e^2 + I*d*h*e^3)*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*(f*x+e)))^2/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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*(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)*(h + I*x)^3), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^2}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________