\(\int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log (e (a+b x)^n (c+d x)^{-n})} \, dx\) [240]

Optimal result

Integrand size = 40, antiderivative size = 88 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{n}\right )}{(b c-a d) n} \] Output:

(b*x+a)^(1+m)*(d*x+c)^(-1-m)*Ei((1+m)*ln(e*(b*x+a)^n/((d*x+c)^n))/n)/(-a*d 
+b*c)/n/((e*(b*x+a)^n/((d*x+c)^n))^((1+m)/n))
 

Mathematica [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx \] Input:

Integrate[((a + b*x)^m*(c + d*x)^(-2 - m))/Log[(e*(a + b*x)^n)/(c + d*x)^n 
],x]
 

Output:

Integrate[((a + b*x)^m*(c + d*x)^(-2 - m))/Log[(e*(a + b*x)^n)/(c + d*x)^n 
], x]
 

Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2973, 2963, 2747, 2609}

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.

\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{-m-2}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx\)

\(\Big \downarrow \) 2973

\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{-m-2}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}dx\)

\(\Big \downarrow \) 2963

\(\displaystyle \frac {(a+b x)^m (c+d x)^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \int \frac {\left (\frac {a+b x}{c+d x}\right )^m}{\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{b c-a d}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} (c+d x)}{a+b x}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+1) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (b c-a d)}\)

Input:

Int[((a + b*x)^m*(c + d*x)^(-2 - m))/Log[(e*(a + b*x)^n)/(c + d*x)^n],x]
 

Output:

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*ExpIntegralEi[((1 + m)*Log[e*((a + b 
*x)/(c + d*x))^n])/n])/((b*c - a*d)*n*(e*((a + b*x)/(c + d*x))^n)^((1 + m) 
/n))
 

Defintions of rubi rules used

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n))   Subst[Int[E^(((m + 1)/n 
)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
 

rule 2963

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[d^2*((g*((a + b*x)/b))^m/(i^2*(b*c - a*d)*(i*((c + d*x)/d))^m*((a 
 + b*x)/(c + d*x))^m))   Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b* 
x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x 
] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + 
 q + 2, 0]
 

rule 2973

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] 
 :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr 
eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !Intege 
rQ[n]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-2-m}}{\ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )}d x\]

Input:

int((b*x+a)^m*(d*x+c)^(-2-m)/ln(e*(b*x+a)^n/((d*x+c)^n)),x)
 

Output:

int((b*x+a)^m*(d*x+c)^(-2-m)/ln(e*(b*x+a)^n/((d*x+c)^n)),x)
 

Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-2-m)/log(e*(b*x+a)^n/((d*x+c)^n)),x, algorit 
hm="fricas")
 

Output:

integral((b*x + a)^m*(d*x + c)^(-m - 2)/log((b*x + a)^n*e/(d*x + c)^n), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**m*(d*x+c)**(-2-m)/ln(e*(b*x+a)**n/((d*x+c)**n)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-2-m)/log(e*(b*x+a)^n/((d*x+c)^n)),x, algorit 
hm="maxima")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m - 2)/log((b*x + a)^n*e/(d*x + c)^n), x 
)
 

Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-2-m)/log(e*(b*x+a)^n/((d*x+c)^n)),x, algorit 
hm="giac")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m - 2)/log((b*x + a)^n*e/(d*x + c)^n), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,{\left (c+d\,x\right )}^{m+2}} \,d x \] Input:

int((a + b*x)^m/(log((e*(a + b*x)^n)/(c + d*x)^n)*(c + d*x)^(m + 2)),x)
 

Output:

int((a + b*x)^m/(log((e*(a + b*x)^n)/(c + d*x)^n)*(c + d*x)^(m + 2)), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx=\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) c^{2}+2 \left (d x +c \right )^{m} \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) c d x +\left (d x +c \right )^{m} \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) d^{2} x^{2}}d x \] Input:

int((b*x+a)^m*(d*x+c)^(-2-m)/log(e*(b*x+a)^n/((d*x+c)^n)),x)
 

Output:

int((a + b*x)**m/((c + d*x)**m*log(((a + b*x)**n*e)/(c + d*x)**n)*c**2 + 2 
*(c + d*x)**m*log(((a + b*x)**n*e)/(c + d*x)**n)*c*d*x + (c + d*x)**m*log( 
((a + b*x)**n*e)/(c + d*x)**n)*d**2*x**2),x)