\(\int \frac {f+g x}{(a+b \log (c (d+e x)^n))^3} \, dx\) [101]

Optimal result

Integrand size = 22, antiderivative size = 261 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac {2 e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \] Output:

1/2*(-d*g+e*f)*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^2/exp(a/b/n)/n^ 
3/((c*(e*x+d)^n)^(1/n))+2*g*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/ 
e^2/exp(2*a/b/n)/n^3/((c*(e*x+d)^n)^(2/n))-1/2*(e*x+d)*(g*x+f)/b/e/n/(a+b* 
ln(c*(e*x+d)^n))^2+1/2*(-d*g+e*f)*(e*x+d)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n) 
)-(e*x+d)*(g*x+f)/b^2/e/n^2/(a+b*ln(c*(e*x+d)^n))
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.98 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-4 g (d+e x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {2 a}{b n}} n \left (c (d+e x)^n\right )^{2/n} \left (b e n (f+g x)+a (e f+d g+2 e g x)+b (d g+e (f+2 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^2 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \] Input:

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

Output:

-1/2*((d + e*x)*(-(E^(a/(b*n))*(e*f - d*g)*(c*(d + e*x)^n)^n^(-1)*ExpInteg 
ralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2) - 4* 
g*(d + e*x)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log 
[c*(d + e*x)^n])^2 + b*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)*(b*e*n*(f + 
 g*x) + a*(e*f + d*g + 2*e*g*x) + b*(d*g + e*(f + 2*g*x))*Log[c*(d + e*x)^ 
n])))/(b^3*e^2*E^((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)*(a + b*Log[c*(d + 
 e*x)^n])^2)
 

Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.59, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2847, 2836, 2734, 2737, 2609, 2847, 2836, 2737, 2609, 2846, 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.

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

\(\Big \downarrow \) 2847

\(\displaystyle -\frac {(e f-d g) \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{2 b e n}+\frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{b n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2836

\(\displaystyle -\frac {(e f-d g) \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}d(d+e x)}{2 b e^2 n}+\frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{b n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2734

\(\displaystyle -\frac {(e f-d g) \left (\frac {\int \frac {1}{a+b \log \left (c (d+e x)^n\right )}d(d+e x)}{b n}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}+\frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{b n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2737

\(\displaystyle -\frac {(e f-d g) \left (\frac {(d+e x) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{a+b \log \left (c (d+e x)^n\right )}d\log \left (c (d+e x)^n\right )}{b n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}+\frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{b n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}dx}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2847

\(\displaystyle \frac {-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )}d(d+e x)}{b e^2 n}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {-\frac {(d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{a+b \log \left (c (d+e x)^n\right )}d\log \left (c (d+e x)^n\right )}{b e^2 n^2}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2846

\(\displaystyle \frac {\frac {2 \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b n}-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 n^2}-\frac {d+e x}{b n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b e^2 n}+\frac {-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 \left (\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n}\right )}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}}{b n}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

Input:

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

Output:

-1/2*((d + e*x)*(f + g*x))/(b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - ((e*f - 
d*g)*(((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*E^( 
a/(b*n))*n^2*(c*(d + e*x)^n)^n^(-1)) - (d + e*x)/(b*n*(a + b*Log[c*(d + e* 
x)^n]))))/(2*b*e^2*n) + (-(((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log 
[c*(d + e*x)^n])/(b*n)])/(b^2*e^2*E^(a/(b*n))*n^2*(c*(d + e*x)^n)^n^(-1))) 
 + (2*(((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b* 
n)])/(b*e^2*E^(a/(b*n))*n*(c*(d + e*x)^n)^n^(-1)) + (g*(d + e*x)^2*ExpInte 
gralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^2*E^((2*a)/(b*n))*n*(c* 
(d + e*x)^n)^(2/n))))/(b*n) - ((d + e*x)*(f + g*x))/(b*e*n*(a + b*Log[c*(d 
 + e*x)^n])))/(b*n)
 

Defintions of rubi rules used

rule 2009

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

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 2734

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b 
*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[(a + b 
*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int 
egerQ[2*p]
 

rule 2737

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

rule 2836

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

rule 2846

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) 
]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* 
x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & 
& IGtQ[q, 0]
 

rule 2847

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e 
*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[( 
f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) 
/(b*e*n*(p + 1)))   Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt 
Q[p, -1] && GtQ[q, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.65 (sec) , antiderivative size = 3114, normalized size of antiderivative = 11.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(3114\)

Input:

int((g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)
 

Output:

-(4*a*e^2*g*x^2+2*a*e^2*f*x+2*b*d^2*g*ln((e*x+d)^n)+3*I*Pi*b*d*e*g*x*csgn( 
I*c*(e*x+d)^n)^2*csgn(I*c)+6*a*d*e*g*x+2*b*d*e*f*n+6*ln(c)*b*d*e*g*x+4*b*e 
^2*g*x^2*ln((e*x+d)^n)+2*b*e^2*f*x*ln((e*x+d)^n)-2*I*Pi*b*e^2*g*x^2*csgn(I 
*c*(e*x+d)^n)^3-I*Pi*b*e^2*f*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn( 
I*c)+2*a*d*e*f+2*ln(c)*b*d^2*g+2*I*Pi*b*e^2*g*x^2*csgn(I*c*(e*x+d)^n)^2*cs 
gn(I*c)-3*I*Pi*b*d*e*g*x*csgn(I*c*(e*x+d)^n)^3-I*Pi*b*d*e*f*csgn(I*c*(e*x+ 
d)^n)^3+6*b*d*e*g*x*ln((e*x+d)^n)+2*b*e^2*g*n*x^2+2*b*e^2*f*n*x+2*ln(c)*b* 
d*e*f+2*b*d*e*f*ln((e*x+d)^n)+2*I*Pi*b*e^2*g*x^2*csgn(I*(e*x+d)^n)*csgn(I* 
c*(e*x+d)^n)^2-I*Pi*b*d^2*g*csgn(I*c*(e*x+d)^n)^3+I*Pi*b*d^2*g*csgn(I*(e*x 
+d)^n)*csgn(I*c*(e*x+d)^n)^2+4*ln(c)*b*e^2*g*x^2+2*ln(c)*b*e^2*f*x-I*Pi*b* 
e^2*f*x*csgn(I*c*(e*x+d)^n)^3+2*b*d*e*g*n*x+I*Pi*b*d^2*g*csgn(I*c*(e*x+d)^ 
n)^2*csgn(I*c)+I*Pi*b*e^2*f*x*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+I*Pi*b*d*e*f 
*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+I*Pi*b*e^2*f*x*csgn(I*(e*x+d)^n)*csgn(I*c 
*(e*x+d)^n)^2-I*Pi*b*d^2*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c) 
-3*I*Pi*b*d*e*g*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)+I*Pi*b*d 
*e*f*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-2*I*Pi*b*e^2*g*x^2*csgn(I*(e* 
x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*Pi*b*d*e*f*csgn(I*(e*x+d)^n)*csgn( 
I*c*(e*x+d)^n)*csgn(I*c)+3*I*Pi*b*d*e*g*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+ 
d)^n)^2+2*a*d^2*g)/(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi* 
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (255) = 510\).

Time = 0.11 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.25 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {{\left ({\left ({\left (b^{2} e f - b^{2} d g\right )} n^{2} \log \left (e x + d\right )^{2} + a^{2} e f - a^{2} d g + {\left (b^{2} e f - b^{2} d g\right )} \log \left (c\right )^{2} + 2 \, {\left ({\left (b^{2} e f - b^{2} d g\right )} n \log \left (c\right ) + {\left (a b e f - a b d g\right )} n\right )} \log \left (e x + d\right ) + 2 \, {\left (a b e f - a b d g\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) - {\left (b^{2} d e f n^{2} + {\left (b^{2} e^{2} g n^{2} + 2 \, a b e^{2} g n\right )} x^{2} + {\left (a b d e f + a b d^{2} g\right )} n + {\left ({\left (b^{2} e^{2} f + b^{2} d e g\right )} n^{2} + {\left (a b e^{2} f + 3 \, a b d e g\right )} n\right )} x + {\left (2 \, b^{2} e^{2} g n^{2} x^{2} + {\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n^{2} x + {\left (b^{2} d e f + b^{2} d^{2} g\right )} n^{2}\right )} \log \left (e x + d\right ) + {\left (2 \, b^{2} e^{2} g n x^{2} + {\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n x + {\left (b^{2} d e f + b^{2} d^{2} g\right )} n\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 4 \, {\left (b^{2} g n^{2} \log \left (e x + d\right )^{2} + b^{2} g \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) + a^{2} g + 2 \, {\left (b^{2} g n \log \left (c\right ) + a b g n\right )} \log \left (e x + d\right )\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} e^{2} n^{5} \log \left (e x + d\right )^{2} + b^{5} e^{2} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e^{2} n^{3} \log \left (c\right ) + a^{2} b^{3} e^{2} n^{3} + 2 \, {\left (b^{5} e^{2} n^{4} \log \left (c\right ) + a b^{4} e^{2} n^{4}\right )} \log \left (e x + d\right )\right )}} \] Input:

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

Output:

1/2*(((b^2*e*f - b^2*d*g)*n^2*log(e*x + d)^2 + a^2*e*f - a^2*d*g + (b^2*e* 
f - b^2*d*g)*log(c)^2 + 2*((b^2*e*f - b^2*d*g)*n*log(c) + (a*b*e*f - a*b*d 
*g)*n)*log(e*x + d) + 2*(a*b*e*f - a*b*d*g)*log(c))*e^((b*log(c) + a)/(b*n 
))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - (b^2*d*e*f*n^2 + (b^ 
2*e^2*g*n^2 + 2*a*b*e^2*g*n)*x^2 + (a*b*d*e*f + a*b*d^2*g)*n + ((b^2*e^2*f 
 + b^2*d*e*g)*n^2 + (a*b*e^2*f + 3*a*b*d*e*g)*n)*x + (2*b^2*e^2*g*n^2*x^2 
+ (b^2*e^2*f + 3*b^2*d*e*g)*n^2*x + (b^2*d*e*f + b^2*d^2*g)*n^2)*log(e*x + 
 d) + (2*b^2*e^2*g*n*x^2 + (b^2*e^2*f + 3*b^2*d*e*g)*n*x + (b^2*d*e*f + b^ 
2*d^2*g)*n)*log(c))*e^(2*(b*log(c) + a)/(b*n)) + 4*(b^2*g*n^2*log(e*x + d) 
^2 + b^2*g*log(c)^2 + 2*a*b*g*log(c) + a^2*g + 2*(b^2*g*n*log(c) + a*b*g*n 
)*log(e*x + d))*log_integral((e^2*x^2 + 2*d*e*x + d^2)*e^(2*(b*log(c) + a) 
/(b*n))))*e^(-2*(b*log(c) + a)/(b*n))/(b^5*e^2*n^5*log(e*x + d)^2 + b^5*e^ 
2*n^3*log(c)^2 + 2*a*b^4*e^2*n^3*log(c) + a^2*b^3*e^2*n^3 + 2*(b^5*e^2*n^4 
*log(c) + a*b^4*e^2*n^4)*log(e*x + d))
 

Sympy [F]

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

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

Output:

Integral((f + g*x)/(a + b*log(c*(d + e*x)**n))**3, x)
 

Maxima [F]

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

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

Output:

-1/2*((2*a*e^2*g + (e^2*g*n + 2*e^2*g*log(c))*b)*x^2 + (d*e*f + d^2*g)*a + 
 (d*e*f*n + (d*e*f + d^2*g)*log(c))*b + ((e^2*f + 3*d*e*g)*a + (e^2*f*n + 
d*e*g*n + (e^2*f + 3*d*e*g)*log(c))*b)*x + (2*b*e^2*g*x^2 + (e^2*f + 3*d*e 
*g)*b*x + (d*e*f + d^2*g)*b)*log((e*x + d)^n))/(b^4*e^2*n^2*log((e*x + d)^ 
n)^2 + b^4*e^2*n^2*log(c)^2 + 2*a*b^3*e^2*n^2*log(c) + a^2*b^2*e^2*n^2 + 2 
*(b^4*e^2*n^2*log(c) + a*b^3*e^2*n^2)*log((e*x + d)^n)) + integrate(1/2*(4 
*e*g*x + e*f + 3*d*g)/(b^3*e*n^2*log((e*x + d)^n) + b^3*e*n^2*log(c) + a*b 
^2*e*n^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4112 vs. \(2 (255) = 510\).

Time = 0.18 (sec) , antiderivative size = 4112, normalized size of antiderivative = 15.75 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*b^2*e*f*n^2*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x 
 + d)^2/((b^5*e^2*n^5*log(e*x + d)^2 + 2*b^5*e^2*n^4*log(e*x + d)*log(c) + 
 2*a*b^4*e^2*n^4*log(e*x + d) + b^5*e^2*n^3*log(c)^2 + 2*a*b^4*e^2*n^3*log 
(c) + a^2*b^3*e^2*n^3)*c^(1/n)) - 1/2*b^2*d*g*n^2*Ei(log(c)/n + a/(b*n) + 
log(e*x + d))*e^(-a/(b*n))*log(e*x + d)^2/((b^5*e^2*n^5*log(e*x + d)^2 + 2 
*b^5*e^2*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^2*n^4*log(e*x + d) + b^5*e^2* 
n^3*log(c)^2 + 2*a*b^4*e^2*n^3*log(c) + a^2*b^3*e^2*n^3)*c^(1/n)) - 1/2*(e 
*x + d)*b^2*e*f*n^2*log(e*x + d)/(b^5*e^2*n^5*log(e*x + d)^2 + 2*b^5*e^2*n 
^4*log(e*x + d)*log(c) + 2*a*b^4*e^2*n^4*log(e*x + d) + b^5*e^2*n^3*log(c) 
^2 + 2*a*b^4*e^2*n^3*log(c) + a^2*b^3*e^2*n^3) - (e*x + d)^2*b^2*g*n^2*log 
(e*x + d)/(b^5*e^2*n^5*log(e*x + d)^2 + 2*b^5*e^2*n^4*log(e*x + d)*log(c) 
+ 2*a*b^4*e^2*n^4*log(e*x + d) + b^5*e^2*n^3*log(c)^2 + 2*a*b^4*e^2*n^3*lo 
g(c) + a^2*b^3*e^2*n^3) + 1/2*(e*x + d)*b^2*d*g*n^2*log(e*x + d)/(b^5*e^2* 
n^5*log(e*x + d)^2 + 2*b^5*e^2*n^4*log(e*x + d)*log(c) + 2*a*b^4*e^2*n^4*l 
og(e*x + d) + b^5*e^2*n^3*log(c)^2 + 2*a*b^4*e^2*n^3*log(c) + a^2*b^3*e^2* 
n^3) + 2*b^2*g*n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(e*x + d))*e^(-2*a/(b* 
n))*log(e*x + d)^2/((b^5*e^2*n^5*log(e*x + d)^2 + 2*b^5*e^2*n^4*log(e*x + 
d)*log(c) + 2*a*b^4*e^2*n^4*log(e*x + d) + b^5*e^2*n^3*log(c)^2 + 2*a*b^4* 
e^2*n^3*log(c) + a^2*b^3*e^2*n^3)*c^(2/n)) + b^2*e*f*n*Ei(log(c)/n + a/(b* 
n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d)*log(c)/((b^5*e^2*n^5*log(e...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(x**2/(log((d + e*x)**n*c)**3*b**3*d + log((d + e*x)**n*c)**3*b**3*e 
*x + 3*log((d + e*x)**n*c)**2*a*b**2*d + 3*log((d + e*x)**n*c)**2*a*b**2*e 
*x + 3*log((d + e*x)**n*c)*a**2*b*d + 3*log((d + e*x)**n*c)*a**2*b*e*x + a 
**3*d + a**3*e*x),x)*log((d + e*x)**n*c)**2*b**3*e**2*g*n + 4*int(x**2/(lo 
g((d + e*x)**n*c)**3*b**3*d + log((d + e*x)**n*c)**3*b**3*e*x + 3*log((d + 
 e*x)**n*c)**2*a*b**2*d + 3*log((d + e*x)**n*c)**2*a*b**2*e*x + 3*log((d + 
 e*x)**n*c)*a**2*b*d + 3*log((d + e*x)**n*c)*a**2*b*e*x + a**3*d + a**3*e* 
x),x)*log((d + e*x)**n*c)*a*b**2*e**2*g*n + 2*int(x**2/(log((d + e*x)**n*c 
)**3*b**3*d + log((d + e*x)**n*c)**3*b**3*e*x + 3*log((d + e*x)**n*c)**2*a 
*b**2*d + 3*log((d + e*x)**n*c)**2*a*b**2*e*x + 3*log((d + e*x)**n*c)*a**2 
*b*d + 3*log((d + e*x)**n*c)*a**2*b*e*x + a**3*d + a**3*e*x),x)*a**2*b*e** 
2*g*n + 2*int(x/(log((d + e*x)**n*c)**3*b**3*d + log((d + e*x)**n*c)**3*b* 
*3*e*x + 3*log((d + e*x)**n*c)**2*a*b**2*d + 3*log((d + e*x)**n*c)**2*a*b* 
*2*e*x + 3*log((d + e*x)**n*c)*a**2*b*d + 3*log((d + e*x)**n*c)*a**2*b*e*x 
 + a**3*d + a**3*e*x),x)*log((d + e*x)**n*c)**2*b**3*d*e*g*n + 2*int(x/(lo 
g((d + e*x)**n*c)**3*b**3*d + log((d + e*x)**n*c)**3*b**3*e*x + 3*log((d + 
 e*x)**n*c)**2*a*b**2*d + 3*log((d + e*x)**n*c)**2*a*b**2*e*x + 3*log((d + 
 e*x)**n*c)*a**2*b*d + 3*log((d + e*x)**n*c)*a**2*b*e*x + a**3*d + a**3*e* 
x),x)*log((d + e*x)**n*c)**2*b**3*e**2*f*n + 4*int(x/(log((d + e*x)**n*c)* 
*3*b**3*d + log((d + e*x)**n*c)**3*b**3*e*x + 3*log((d + e*x)**n*c)**2*...