\(\int \frac {(a+b x)^3}{(c+d x)^5 \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [241]

Optimal result

Integrand size = 35, antiderivative size = 75 \[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^4} \] Output:

(b*x+a)^4*Ei(4*ln(e*((b*x+a)/(d*x+c))^n)/n)/(-a*d+b*c)/n/((e*((b*x+a)/(d*x 
+c))^n)^(4/n))/(d*x+c)^4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^4} \] Input:

Integrate[(a + b*x)^3/((c + d*x)^5*Log[e*((a + b*x)/(c + d*x))^n]),x]
 

Output:

((a + b*x)^4*ExpIntegralEi[(4*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - 
a*d)*n*(e*((a + b*x)/(c + d*x))^n)^(4/n)*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2961, 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.

\(\displaystyle \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\)

\(\Big \downarrow \) 2961

\(\displaystyle \frac {\int \frac {(a+b x)^3}{(c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}d\frac {a+b x}{c+d x}}{b c-a d}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \int \frac {\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{4/n} (c+d x)}{a+b x}d\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (c+d x)^4 (b c-a d)}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^4 (b c-a d)}\)

Input:

Int[(a + b*x)^3/((c + d*x)^5*Log[e*((a + b*x)/(c + d*x))^n]),x]
 

Output:

((a + b*x)^4*ExpIntegralEi[(4*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - 
a*d)*n*(e*((a + b*x)/(c + d*x))^n)^(4/n)*(c + d*x)^4)
 

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{3}}{\left (d x +c \right )^{5} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]

Input:

int((b*x+a)^3/(d*x+c)^5/ln(e*((b*x+a)/(d*x+c))^n),x)
 

Output:

int((b*x+a)^3/(d*x+c)^5/ln(e*((b*x+a)/(d*x+c))^n),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\frac {\operatorname {log\_integral}\left (\frac {{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} e^{\frac {4}{n}}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\right )}{{\left (b c - a d\right )} e^{\frac {4}{n}} n} \] Input:

integrate((b*x+a)^3/(d*x+c)^5/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fri 
cas")
 

Output:

log_integral((b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*e^( 
4/n)/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4))/((b*c - a* 
d)*e^(4/n)*n)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**3/(d*x+c)**5/ln(e*((b*x+a)/(d*x+c))**n),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{{\left (d x + c\right )}^{5} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:

integrate((b*x+a)^3/(d*x+c)^5/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="max 
ima")
 

Output:

integrate((b*x + a)^3/((d*x + c)^5*log(e*((b*x + a)/(d*x + c))^n)), x)
 

Giac [F]

\[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{{\left (d x + c\right )}^{5} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )} \,d x } \] Input:

integrate((b*x+a)^3/(d*x+c)^5/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="gia 
c")
 

Output:

integrate((b*x + a)^3/((d*x + c)^5*log(e*((b*x + a)/(d*x + c))^n)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {{\left (a+b\,x\right )}^3}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (c+d\,x\right )}^5} \,d x \] Input:

int((a + b*x)^3/(log(e*((a + b*x)/(c + d*x))^n)*(c + d*x)^5),x)
 

Output:

int((a + b*x)^3/(log(e*((a + b*x)/(c + d*x))^n)*(c + d*x)^5), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {too large to display} \] Input:

int((b*x+a)^3/(d*x+c)^5/log(e*((b*x+a)/(d*x+c))^n),x)
 

Output:

(4*int(x**3/(log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**5 + 5*log(((a + b*x)* 
*n*e)/(c + d*x)**n)*a*c**4*d*x + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c 
**3*d**2*x**2 + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**2*d**3*x**3 + 5 
*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c*d**4*x**4 + log(((a + b*x)**n*e)/( 
c + d*x)**n)*a*d**5*x**5 + log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**5*x + 5 
*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**4*d*x**2 + 10*log(((a + b*x)**n*e 
)/(c + d*x)**n)*b*c**3*d**2*x**3 + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*b 
*c**2*d**3*x**4 + 5*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c*d**4*x**5 + log 
(((a + b*x)**n*e)/(c + d*x)**n)*b*d**5*x**6),x)*a**2*b**3*d**5*n - 8*int(x 
**3/(log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**5 + 5*log(((a + b*x)**n*e)/(c 
 + d*x)**n)*a*c**4*d*x + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**3*d**2 
*x**2 + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**2*d**3*x**3 + 5*log(((a 
 + b*x)**n*e)/(c + d*x)**n)*a*c*d**4*x**4 + log(((a + b*x)**n*e)/(c + d*x) 
**n)*a*d**5*x**5 + log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**5*x + 5*log(((a 
 + b*x)**n*e)/(c + d*x)**n)*b*c**4*d*x**2 + 10*log(((a + b*x)**n*e)/(c + d 
*x)**n)*b*c**3*d**2*x**3 + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c**2*d* 
*3*x**4 + 5*log(((a + b*x)**n*e)/(c + d*x)**n)*b*c*d**4*x**5 + log(((a + b 
*x)**n*e)/(c + d*x)**n)*b*d**5*x**6),x)*a*b**4*c*d**4*n + 4*int(x**3/(log( 
((a + b*x)**n*e)/(c + d*x)**n)*a*c**5 + 5*log(((a + b*x)**n*e)/(c + d*x)** 
n)*a*c**4*d*x + 10*log(((a + b*x)**n*e)/(c + d*x)**n)*a*c**3*d**2*x**2 ...