3.6.29 \(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{(g+h x) (i+j x)^3} \, dx\) [529]

3.6.29.1 Optimal result

Integrand size = 33, antiderivative size = 425 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=-\frac {b f p q}{2 (f i-e j) (h i-g j) (i+j x)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (h i-g j) (i+j x)^2}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2 (i+j x)}+\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{(h i-g j)^3}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}-\frac {h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{(h i-g j)^3}+\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3} \]

output

-1/2*b*f*p*q/(-e*j+f*i)/(-g*j+h*i)/(j*x+i)-b*f*h*p*q*ln(f*x+e)/(-e*j+f*i)/ 
(-g*j+h*i)^2-1/2*b*f^2*p*q*ln(f*x+e)/(-e*j+f*i)^2/(-g*j+h*i)+1/2*(a+b*ln(c 
*(d*(f*x+e)^p)^q))/(-g*j+h*i)/(j*x+i)^2+h*(a+b*ln(c*(d*(f*x+e)^p)^q))/(-g* 
j+h*i)^2/(j*x+i)+h^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*(h*x+g)/(-e*h+f*g))/ 
(-g*j+h*i)^3+b*f*h*p*q*ln(j*x+i)/(-e*j+f*i)/(-g*j+h*i)^2+1/2*b*f^2*p*q*ln( 
j*x+i)/(-e*j+f*i)^2/(-g*j+h*i)-h^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*(j*x+i 
)/(-e*j+f*i))/(-g*j+h*i)^3+b*h^2*p*q*polylog(2,-h*(f*x+e)/(-e*h+f*g))/(-g* 
j+h*i)^3-b*h^2*p*q*polylog(2,-j*(f*x+e)/(-e*j+f*i))/(-g*j+h*i)^3
 

3.6.29.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\frac {\frac {a (h i-g j)^2}{(i+j x)^2}+\frac {2 a h (h i-g j)}{i+j x}+\frac {b (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{(i+j x)^2}+\frac {2 b h (h i-g j) \log \left (c \left (d (e+f x)^p\right )^q\right )}{i+j x}+2 h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-\frac {2 b f h (h i-g j) p q (\log (e+f x)-\log (i+j x))}{f i-e j}-\frac {b f (h i-g j)^2 p q (f i-e j+f (i+j x) \log (e+f x)-f (i+j x) \log (i+j x))}{(f i-e j)^2 (i+j x)}-2 h^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )+2 b h^2 p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-2 b h^2 p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{2 (h i-g j)^3} \]

input

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/((g + h*x)*(i + j*x)^3),x]
 

output

((a*(h*i - g*j)^2)/(i + j*x)^2 + (2*a*h*(h*i - g*j))/(i + j*x) + (b*(h*i - 
 g*j)^2*Log[c*(d*(e + f*x)^p)^q])/(i + j*x)^2 + (2*b*h*(h*i - g*j)*Log[c*( 
d*(e + f*x)^p)^q])/(i + j*x) + 2*h^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[ 
(f*(g + h*x))/(f*g - e*h)] - (2*b*f*h*(h*i - g*j)*p*q*(Log[e + f*x] - Log[ 
i + j*x]))/(f*i - e*j) - (b*f*(h*i - g*j)^2*p*q*(f*i - e*j + f*(i + j*x)*L 
og[e + f*x] - f*(i + j*x)*Log[i + j*x]))/((f*i - e*j)^2*(i + j*x)) - 2*h^2 
*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)] + 2*b*h^2 
*p*q*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 2*b*h^2*p*q*PolyLog[2, (j* 
(e + f*x))/(-(f*i) + e*j)])/(2*(h*i - g*j)^3)
 

3.6.29.3 Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2895, 2865, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3}dx\)

\(\Big \downarrow \) 2865

\(\displaystyle \int \left (\frac {h^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(g+h x) (h i-g j)^3}-\frac {h^2 j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^3}-\frac {h j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x)^2 (h i-g j)^2}-\frac {j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x)^3 (h i-g j)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {h^2 \log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}-\frac {h^2 \log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^3}+\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)^2}+\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{2 (i+j x)^2 (h i-g j)}-\frac {b f^2 p q \log (e+f x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b f^2 p q \log (i+j x)}{2 (f i-e j)^2 (h i-g j)}+\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{(h i-g j)^3}-\frac {b h^2 p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{(h i-g j)^3}-\frac {b f p q}{2 (i+j x) (f i-e j) (h i-g j)}-\frac {b f h p q \log (e+f x)}{(f i-e j) (h i-g j)^2}+\frac {b f h p q \log (i+j x)}{(f i-e j) (h i-g j)^2}\)

input

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/((g + h*x)*(i + j*x)^3),x]
 

output

-1/2*(b*f*p*q)/((f*i - e*j)*(h*i - g*j)*(i + j*x)) - (b*f*h*p*q*Log[e + f* 
x])/((f*i - e*j)*(h*i - g*j)^2) - (b*f^2*p*q*Log[e + f*x])/(2*(f*i - e*j)^ 
2*(h*i - g*j)) + (a + b*Log[c*(d*(e + f*x)^p)^q])/(2*(h*i - g*j)*(i + j*x) 
^2) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q]))/((h*i - g*j)^2*(i + j*x)) + (h^ 
2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - 
g*j)^3 + (b*f*h*p*q*Log[i + j*x])/((f*i - e*j)*(h*i - g*j)^2) + (b*f^2*p*q 
*Log[i + j*x])/(2*(f*i - e*j)^2*(h*i - g*j)) - (h^2*(a + b*Log[c*(d*(e + f 
*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^3 + (b*h^2*p*q*Poly 
Log[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^3 - (b*h^2*p*q*PolyLog[2 
, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^3
 

3.6.29.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> 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] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

rule 2895

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]]
 

3.6.29.4 Maple [F]

\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )^{3}}d x\]

input

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x)
 

output

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x)
 

3.6.29.5 Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="fri 
cas")
 

output

integral((b*log(((f*x + e)^p*d)^q*c) + a)/(h*j^3*x^4 + g*i^3 + (3*h*i*j^2 
+ g*j^3)*x^3 + 3*(h*i^2*j + g*i*j^2)*x^2 + (h*i^3 + 3*g*i^2*j)*x), x)
 

3.6.29.6 Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )^{3}}\, dx \]

input

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g)/(j*x+i)**3,x)
 

output

Integral((a + b*log(c*(d*(e + f*x)**p)**q))/((g + h*x)*(i + j*x)**3), x)
 

3.6.29.7 Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="max 
ima")
 

output

1/2*(2*h^2*log(h*x + g)/(h^3*i^3 - 3*g*h^2*i^2*j + 3*g^2*h*i*j^2 - g^3*j^3 
) - 2*h^2*log(j*x + i)/(h^3*i^3 - 3*g*h^2*i^2*j + 3*g^2*h*i*j^2 - g^3*j^3) 
 + (2*h*j*x + 3*h*i - g*j)/(h^2*i^4 - 2*g*h*i^3*j + g^2*i^2*j^2 + (h^2*i^2 
*j^2 - 2*g*h*i*j^3 + g^2*j^4)*x^2 + 2*(h^2*i^3*j - 2*g*h*i^2*j^2 + g^2*i*j 
^3)*x))*a + b*integrate((q*log(d) + log(((f*x + e)^p)^q) + log(c))/(h*j^3* 
x^4 + g*i^3 + (3*h*i*j^2 + g*j^3)*x^3 + 3*(h*i^2*j + g*i*j^2)*x^2 + (h*i^3 
 + 3*g*i^2*j)*x), x)
 

3.6.29.8 Giac [F]

\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}^{3}} \,d x } \]

input

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)/(j*x+i)^3,x, algorithm="gia 
c")
 

output

integrate((b*log(((f*x + e)^p*d)^q*c) + a)/((h*x + g)*(j*x + i)^3), x)
 

3.6.29.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,{\left (i+j\,x\right )}^3} \,d x \]

input

int((a + b*log(c*(d*(e + f*x)^p)^q))/((g + h*x)*(i + j*x)^3),x)
 

output

int((a + b*log(c*(d*(e + f*x)^p)^q))/((g + h*x)*(i + j*x)^3), x)