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

Optimal. Leaf size=172 \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\log (g k+h k x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h k}-\frac{p r \log (g k+h k x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h k}-\frac{q r \log (g k+h k x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h k} \]

[Out]

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

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Rubi [A]  time = 0.135941, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2494, 2394, 2393, 2391} \[ -\frac{p r \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{h k}-\frac{q r \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{h k}+\frac{\log (g k+h k x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h k}-\frac{p r \log (g k+h k x) \log \left (-\frac{h (a+b x)}{b g-a h}\right )}{h k}-\frac{q r \log (g k+h k x) \log \left (-\frac{h (c+d x)}{d g-c h}\right )}{h k} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2494

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]

Rule 2394

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 2393

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.0887067, size = 166, normalized size = 0.97 \[ \frac{p r \text{PolyLog}\left (2,\frac{h (a+b x)}{a h-b g}\right )+q r \text{PolyLog}\left (2,\frac{h (c+d x)}{c h-d g}\right )+\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log (a+b x) \log (g+h x)+p r \log (a+b x) \log \left (\frac{b (g+h x)}{b g-a h}\right )-q r \log (c+d x) \log (g+h x)+q r \log (c+d x) \log \left (\frac{d (g+h x)}{d g-c h}\right )}{h k} \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 0.491, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{hkx+gk}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.26247, size = 275, normalized size = 1.6 \begin{align*} \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 k} + \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 k}\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 k x + g k\right )}{f h k} + \frac{\log \left (h k x + g k\right ) \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k),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*k) + (log(d*x +
c)*log((d*h*x + c*h)/(d*g - c*h) + 1) + dilog(-(d*h*x + c*h)/(d*g - c*h)))*f*q/(h*k))*r/f - (f*p*log(b*x + a)
+ f*q*log(d*x + c))*r*log(h*k*x + g*k)/(f*h*k) + log(h*k*x + g*k)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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