\(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x) (h+i x)} \, dx\) [221]

Optimal result

Integrand size = 29, antiderivative size = 155 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2465, 2441, 2440, 2438} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]

[In]

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

[Out]

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

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-b n \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )}{g h-f i} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.12 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.44

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (i x +h \right )}{f i -g h}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f i -g h}-\frac {b n \operatorname {dilog}\left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{f i -g h}-\frac {b n \ln \left (i x +h \right ) \ln \left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{f i -g h}+\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f i -g h}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f i -g h}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\ln \left (i x +h \right )}{f i -g h}-\frac {\ln \left (g x +f \right )}{f i -g h}\right )\)	\(378\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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