\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{g+h x} \, dx\) [30]

Optimal result

Integrand size = 29, antiderivative size = 148 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {q r \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2580, 2441, 2440, 2438} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\frac {\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {p r \log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {q r \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {q r \log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h} \]

[In]

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

[Out]

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

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 2580

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[Log[g + h*x]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/h), x] + (-Dist[b*p*(r/h), Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[d*q*(r/h), Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\frac {-p r \log (a+b x) \log (g+h x)-q r \log (c+d x) \log (g+h x)+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+p r \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+q r \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+p r \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{-b g+a h}\right )+q r \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{-d g+c h}\right )}{h} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 18.98 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.26

method	result	size
parts	\(\frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) \ln \left (h x +g \right )}{h}-\frac {r \left (b p h \left (\frac {\operatorname {dilog}\left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{b}+\frac {\ln \left (h x +g \right ) \ln \left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{b}\right )+d q h \left (\frac {\operatorname {dilog}\left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{d}+\frac {\ln \left (h x +g \right ) \ln \left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{d}\right )\right )}{h^{2}}\)	\(186\)

[In]

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

[Out]

ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*ln(h*x+g)/h-1/h^2*r*(b*p*h*(dilog(((h*x+g)*b+a*h-b*g)/(a*h-b*g))/b+ln(h*x+g)*l
n(((h*x+g)*b+a*h-b*g)/(a*h-b*g))/b)+d*q*h*(dilog((d*(h*x+g)+c*h-d*g)/(c*h-d*g))/d+ln(h*x+g)*ln((d*(h*x+g)+c*h-
d*g)/(c*h-d*g))/d))

Fricas [F]

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

[In]

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

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*x + g), x)

Sympy [F(-1)]

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

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*x+g),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.26 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\frac {{\left (\frac {{\left (\log \left (b x + a\right ) \log \left (\frac {b h x + a h}{b g - a h} + 1\right ) + {\rm Li}_2\left (-\frac {b h x + a h}{b g - a h}\right )\right )} f p}{h} + \frac {{\left (\log \left (d x + c\right ) \log \left (\frac {d h x + c h}{d g - c h} + 1\right ) + {\rm Li}_2\left (-\frac {d h x + c h}{d g - c h}\right )\right )} f q}{h}\right )} r}{f} - \frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (h x + g\right )}{f h} + \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (h x + g\right )}{h} \]

[In]

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

[Out]

((log(b*x + a)*log((b*h*x + a*h)/(b*g - a*h) + 1) + dilog(-(b*h*x + a*h)/(b*g - a*h)))*f*p/h + (log(d*x + c)*l
og((d*h*x + c*h)/(d*g - c*h) + 1) + dilog(-(d*h*x + c*h)/(d*g - c*h)))*f*q/h)*r/f - (f*p*log(b*x + a) + f*q*lo
g(d*x + c))*r*log(h*x + g)/(f*h) + log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(h*x + g)/h

Giac [F]

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

[In]

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

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*x + g), x)

Mupad [F(-1)]

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

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/(g + h*x),x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/(g + h*x), x)