\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{(g+h x) (i+j x)} \, dx\) [527]

Optimal result

Integrand size = 33, antiderivative size = 165 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2465, 2441, 2440, 2438, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]

[In]

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

[Out]

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

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]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log \left (\frac {f (g+h x)}{f g-e h}\right )-\log \left (\frac {f (i+j x)}{f i-e j}\right )\right )+b p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-b p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{h i-g j} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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