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 \[ \frac {\log (g k+h k x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h k}-\frac {p r \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {p r \log (g k+h k x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h k}-\frac {q r \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {q r \log (g k+h k x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h k} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 172, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.09, size = 166, normalized size = 0.97 \[ \frac {\log (g+h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \text {Li}_2\left (\frac {h (a+b x)}{a h-b g}\right )-p r \log (a+b x) \log (g+h x)+p r \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+q r \text {Li}_2\left (\frac {h (c+d x)}{c h-d g}\right )-q r \log (c+d x) \log (g+h x)+q r \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

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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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k}\,{d x} \]

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)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{h k x +g k}\, dx \]

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 = 1.20, size = 204, normalized size = 1.19 \[ \frac {{\left (\frac {{\left (\log \left (b x + a\right ) \log \left (\frac {b h x + a h}{b g - a h} + 1\right ) + {\rm Li}_2\left (-\frac {b h x + a h}{b g - a h}\right )\right )} f p}{h k} + \frac {{\left (\log \left (d x + c\right ) \log \left (\frac {d h x + c h}{d g - c h} + 1\right ) + {\rm Li}_2\left (-\frac {d h x + c h}{d g - c h}\right )\right )} f q}{h k}\right )} r}{f} - \frac {{\left (f p \log \left (b x + a\right ) + f q \log \left (d x + c\right )\right )} r \log \left (h k x + g k\right )}{f h k} + \frac {\log \left (h k x + g k\right ) \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{g\,k+h\,k\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{g + h x}\, dx}{k} \]

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]

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

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