\(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x) (h+i x)^2} \, dx\) [222]

Optimal result

Integrand size = 29, antiderivative size = 252 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^2} \, dx=-\frac {b e n \log (d+e x)}{(e h-d i) (g h-f i)}+\frac {a+b \log \left (c (d+e x)^n\right )}{(g h-f i) (h+i x)}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(g h-f i)^2}+\frac {b e n \log (h+i x)}{(e h-d i) (g h-f i)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )}{(g h-f i)^2}+\frac {b g n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {b g n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2465, 2441, 2440, 2438, 2442, 36, 31} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^2} \, dx=\frac {a+b \log \left (c (d+e x)^n\right )}{(h+i x) (g h-f i)}+\frac {g \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}-\frac {g \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2}+\frac {b g n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^2}-\frac {b g n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^2}-\frac {b e n \log (d+e x)}{(e h-d i) (g h-f i)}+\frac {b e n \log (h+i x)}{(e h-d i) (g h-f i)} \]

[In]

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

[Out]

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

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 2442

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)^2} \, dx=\frac {\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{h+i x}+g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-\frac {b e (g h-f i) n (\log (d+e x)-\log (h+i x))}{e h-d i}-g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )+b g n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-b g n \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )}{(g h-f i)^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.72 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.96

method	result	size
risch	\(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{\left (f i -g h \right ) \left (i x +h \right )}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (i x +h \right )}{\left (f i -g h \right )^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (g x +f \right )}{\left (f i -g h \right )^{2}}-\frac {b n g \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{\left (f i -g h \right )^{2}}-\frac {b n g \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{\left (f i -g h \right )^{2}}-\frac {b e n \ln \left (e x +d \right )}{\left (f i -g h \right ) \left (d i -e h \right )}+\frac {b e n \ln \left (i x +h \right )}{\left (f i -g h \right ) \left (d i -e h \right )}+\frac {b n g \operatorname {dilog}\left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{\left (f i -g h \right )^{2}}+\frac {b n g \ln \left (i x +h \right ) \ln \left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{\left (f i -g h \right )^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {1}{\left (f i -g h \right ) \left (i x +h \right )}-\frac {g \ln \left (i x +h \right )}{\left (f i -g h \right )^{2}}+\frac {g \ln \left (g x +f \right )}{\left (f i -g h \right )^{2}}\right )\)	\(494\)

[In]

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

[Out]

-b*ln((e*x+d)^n)/(f*i-g*h)/(i*x+h)-b*ln((e*x+d)^n)*g/(f*i-g*h)^2*ln(i*x+h)+b*ln((e*x+d)^n)*g/(f*i-g*h)^2*ln(g*
x+f)-b*n*g/(f*i-g*h)^2*dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))-b*n*g/(f*i-g*h)^2*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)
/(d*g-e*f))-b*e*n/(f*i-g*h)/(d*i-e*h)*ln(e*x+d)+b*e*n/(f*i-g*h)/(d*i-e*h)*ln(i*x+h)+b*n*g/(f*i-g*h)^2*dilog(((
i*x+h)*e+d*i-e*h)/(d*i-e*h))+b*n*g/(f*i-g*h)^2*ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))+(-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)/(i*x+h)-g/(f*i-g*h)^2*ln(i
*x+h)+g/(f*i-g*h)^2*ln(g*x+f))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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