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

Optimal result

Integrand size = 32, antiderivative size = 485 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx=\frac {b f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {(a+b \log (c (e+f x)))^2}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^3 (h+i x)}-\frac {b^2 f^2 \log (h+i x)}{d (f h-e i)^3}+\frac {2 b f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac {b f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {2 b f^2 (a+b \log (c (e+f x))) \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]

[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] (verified)

Time = 0.66 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2458, 12, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx=\frac {2 b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}+\frac {2 b f^2 \log \left (\frac {f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac {f^2 \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^3}+\frac {b f^2 \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac {f i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^3}+\frac {b f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac {(a+b \log (c (e+f x)))^2}{2 d (h+i x)^2 (f h-e i)}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {b^2 f^2 \log (h+i x)}{d (f h-e i)^3} \]

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

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx=\frac {3 a^2 (f h-e i)^2+6 a^2 f (f h-e i) (h+i x)+6 a^2 f^2 (h+i x)^2 \log (e+f x)-6 a^2 f^2 (h+i x)^2 \log (h+i x)+6 a b \left ((f h-e i)^2 \log (c (e+f x))+f^2 (h+i x)^2 \log ^2(c (e+f x))-f (h+i x) (f h-e i+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))-2 f (h+i x) (f (h+i x) \log (e+f x)+(-f h+e i) \log (c (e+f x))-f (h+i x) \log (h+i x))-2 f^2 (h+i x)^2 \left (\log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+\operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )\right )\right )+b^2 \left (2 f^2 (h+i x)^2 \log ^3(c (e+f x))-6 f (h+i x) \left (\log (c (e+f x)) \left (i (e+f x) \log (c (e+f x))-2 f (h+i x) \log \left (\frac {f (h+i x)}{f h-e i}\right )\right )-2 f (h+i x) \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )\right )+3 \left ((f h-e i)^2 \log ^2(c (e+f x))+f (h+i x) \left (2 f (h+i x) \log (e+f x)-2 (f h-e i) \log (c (e+f x))-f (h+i x) \log ^2(c (e+f x))-2 f (h+i x) \log (h+i x)+2 f (h+i x) \log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+2 f (h+i x) \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )\right )\right )-6 f^2 (h+i x)^2 \left (\log ^2(c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+2 \log (c (e+f x)) \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )-2 \operatorname {PolyLog}\left (3,\frac {i (e+f x)}{-f h+e i}\right )\right )\right )}{6 d (f h-e i)^3 (h+i x)^2} \]

[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*(2*f^2*(h + i*x)^2*Log[c*(e + f*x)]^3 - 6*
f*(h + i*x)*(Log[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)]) + 3*((f*h - e*i)^2*Log[c*(e + f*x)]^2 + f*(h + i*x)*(2
*f*(h + i*x)*Log[e + f*x] - 2*(f*h - e*i)*Log[c*(e + f*x)] - f*(h + i*x)*Log[c*(e + f*x)]^2 - 2*f*(h + i*x)*Lo
g[h + i*x] + 2*f*(h + i*x)*Log[c*(e + f*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]

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

[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)

Fricas [F]

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

[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]

\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx=\frac {\int \frac {a^{2}}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx + \int \frac {b^{2} \log {\left (c e + c f x \right )}^{2}}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx + \int \frac {2 a b \log {\left (c e + c f x \right )}}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx}{d} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1271 vs. \(2 (480) = 960\).

Time = 0.38 (sec) , antiderivative size = 1271, normalized size of antiderivative = 2.62 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^3} \, dx=\text {Too large to display} \]

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)