\(\int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx\) [182]

Optimal result

Integrand size = 30, antiderivative size = 250 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=-\frac {b f}{2 d (f h-e i)^2 (h+i x)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2458, 12, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=-\frac {f^2 \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (h+i x)^2 (f h-e i)}+\frac {b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {b f}{2 d (h+i x) (f h-e i)^2} \]

[In]

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

[Out]

-1/2*(b*f)/(d*(f*h - e*i)^2*(h + i*x)) - (b*f^2*Log[e + f*x])/(2*d*(f*h - e*i)^3) + (a + b*Log[c*(e + f*x)])/(
2*d*(f*h - e*i)*(h + i*x)^2) - (f*i*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*(f*h - e*i)^3*(h + i*x)) + (3*b*f^2
*Log[h + i*x])/(2*d*(f*h - e*i)^3) - (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) + (b*f^2*PolyLog[2, -((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 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 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 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]

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\frac {\frac {(f h-e i)^2 (a+b \log (c (e+f x)))}{(h+i x)^2}+\frac {2 f (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f^2 (a+b \log (c (e+f x)))^2}{b}+2 b f^2 (-\log (e+f x)+\log (h+i x))-\frac {b f (f h-e i+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))}{h+i x}-2 f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f^2 \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(241)=482\).

Time = 1.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.04

method	result	size
parts	\(\frac {a \left (-\frac {1}{2 \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {f^{2} \ln \left (i x +h \right )}{\left (e i -f h \right )^{3}}+\frac {f}{\left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (e i -f h \right )^{3}}\right )}{d}+\frac {b \left (-\frac {c \,f^{3} \ln \left (c f x +c e \right )^{2}}{2 \left (e i -f h \right )^{3}}+\frac {c^{3} f^{3} i \left (-\frac {\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{i}+\frac {c \left (e i -f h \right )}{i \left (-c e i +h c f +i \left (c f x +c e \right )\right )}}{2 c^{2} \left (e i -f h \right )^{2}}+\frac {\ln \left (c f x +c e \right ) \left (-2 c e i +2 h c f +i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 \left (-c e i +h c f +i \left (c f x +c e \right )\right )^{2} c^{2} \left (e i -f h \right )^{2}}\right )}{e i -f h}+\frac {c \,f^{3} i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{\left (e i -f h \right )^{3}}-\frac {c^{2} f^{3} i \left (\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}-\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (-c e i +h c f +i \left (c f x +c e \right )\right )}\right )}{\left (e i -f h \right )^{2}}\right )}{d c f}\)	\(510\)
derivativedivides	\(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\)	\(580\)
default	\(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\)	\(580\)
risch	\(-\frac {a}{2 d \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {a \,f^{2} \ln \left (i x +h \right )}{d \left (e i -f h \right )^{3}}+\frac {a f}{d \left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {a \,f^{2} \ln \left (f x +e \right )}{d \left (e i -f h \right )^{3}}-\frac {b \,f^{2} \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{3}}-\frac {3 b \,f^{2} \ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{2 d \left (e i -f h \right )^{3}}-\frac {b c \,f^{2} i e}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{3} h}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b \,c^{2} f^{4} i^{2} \ln \left (c f x +c e \right ) x^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{4} i \ln \left (c f x +c e \right ) h x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}-\frac {b \,c^{2} f^{2} i^{2} \ln \left (c f x +c e \right ) e^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{3} i \ln \left (c f x +c e \right ) e h}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,f^{2} \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b \,f^{2} \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b c \,f^{3} i \ln \left (c f x +c e \right ) x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{2} i \ln \left (c f x +c e \right ) e}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}\)	\(619\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*log(c*f*x + c*e) + a)/(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]

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

[In]

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

[Out]

(Integral(a/(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*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 [F]

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

[In]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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