\(\int \frac {(h+i x) (a+b \log (c (d+e x)^n))}{f+g x} \, dx\) [219]

Optimal result

Integrand size = 27, antiderivative size = 119 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {a i x}{g}-\frac {b i n x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b (g h-f i) n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2465, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a i x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {b n (g h-f i) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {b i n x}{g} \]

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {a g i x-b g i n x+\frac {b g i (d+e x) \log \left (c (d+e x)^n\right )}{e}+(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+b (g h-f i) n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{g^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.78 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.31

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x i}{g}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) f i}{g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right ) h}{g}-\frac {b i n x}{g}-\frac {b n i f}{g^{2}}+\frac {b n i d \ln \left (\left (g x +f \right ) e +d g -e f \right )}{e g}+\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) f i}{g^{2}}-\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) h}{g}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) f i}{g^{2}}-\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right ) h}{g}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x i}{g}+\frac {\left (-f i +g h \right ) \ln \left (g x +f \right )}{g^{2}}\right )\)	\(394\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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