\(\int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^2} \, dx\) [181]

Optimal result

Integrand size = 30, antiderivative size = 151 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^2} \, dx=-\frac {i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^2 (h+i x)}+\frac {b f \log (h+i x)}{d (f h-e i)^2}-\frac {f (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2458, 12, 2389, 2379, 2438, 2351, 31} \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^2} \, dx=-\frac {f \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}-\frac {i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {b f \log (h+i x)}{d (f h-e i)^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^2} \, dx=\frac {\frac {2 (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f (a+b \log (c (e+f x)))^2}{b}+2 b f (-\log (e+f x)+\log (h+i x))-2 f (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(316\) vs. \(2(150)=300\).

Time = 1.07 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.10

method	result	size
parts	\(\frac {a \left (-\frac {1}{\left (e i -f h \right ) \left (i x +h \right )}-\frac {f \ln \left (i x +h \right )}{\left (e i -f h \right )^{2}}+\frac {f \ln \left (f x +e \right )}{\left (e i -f h \right )^{2}}\right )}{d}+\frac {b \left (\frac {c \,f^{2} \ln \left (c f x +c e \right )^{2}}{2 \left (e i -f h \right )^{2}}-\frac {c \,f^{2} i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{\left (e i -f h \right )^{2}}+\frac {c^{2} f^{2} i \left (\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}-\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (-c e i +h c f +i \left (c f x +c e \right )\right )}\right )}{e i -f h}\right )}{d c f}\)	\(317\)
risch	\(-\frac {a}{d \left (e i -f h \right ) \left (i x +h \right )}-\frac {a f \ln \left (i x +h \right )}{d \left (e i -f h \right )^{2}}+\frac {a f \ln \left (f x +e \right )}{d \left (e i -f h \right )^{2}}+\frac {b f \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{2}}-\frac {b f \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}-\frac {b f \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{2}}+\frac {b f \ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )^{2}}-\frac {b c \,f^{2} i \ln \left (c f x +c e \right ) x}{d \left (e i -f h \right )^{2} \left (c f i x +h c f \right )}-\frac {b c f i \ln \left (c f x +c e \right ) e}{d \left (e i -f h \right )^{2} \left (c f i x +h c f \right )}\)	\(330\)
derivativedivides	\(\frac {\frac {c^{3} f^{2} a \left (\frac {1}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{2} \left (e i -f h \right )^{2}}+\frac {\ln \left (c f x +c e \right )}{c^{2} \left (e i -f h \right )^{2}}\right )}{d}+\frac {c^{3} f^{2} b \left (\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c \left (e i -f h \right )}+\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{2} \left (e i -f h \right )^{2}}\right )}{d}}{c f}\)	\(367\)
default	\(\frac {\frac {c^{3} f^{2} a \left (\frac {1}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{2} \left (e i -f h \right )^{2}}+\frac {\ln \left (c f x +c e \right )}{c^{2} \left (e i -f h \right )^{2}}\right )}{d}+\frac {c^{3} f^{2} b \left (\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c \left (e i -f h \right )}+\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{2} \left (e i -f h \right )^{2}}\right )}{d}}{c f}\)	\(367\)

[In]

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

[Out]

a/d*(-1/(e*i-f*h)/(i*x+h)-f/(e*i-f*h)^2*ln(i*x+h)+f/(e*i-f*h)^2*ln(f*x+e))+b/d/c/f*(1/2*c*f^2/(e*i-f*h)^2*ln(c
*f*x+c*e)^2-c*f^2/(e*i-f*h)^2*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^2/(e*i-f*h)*i*(1/c/(e*i-f*h)*ln(-c*e*i+h*c*f+i*(c*f*x+c*e))/i-l
n(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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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