\(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^3} \, dx\) [100]

Optimal result

Integrand size = 24, antiderivative size = 351 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {(e f-d g) (d+e x) (f+g x)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \] Output:

1/2*(-d*g+e*f)^2*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(a/b/n)/ 
n^3/((c*(e*x+d)^n)^(1/n))+4*g*(-d*g+e*f)*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^ 
n))/b/n)/b^3/e^3/exp(2*a/b/n)/n^3/((c*(e*x+d)^n)^(2/n))+9/2*g^2*(e*x+d)^3* 
Ei(3*(a+b*ln(c*(e*x+d)^n))/b/n)/b^3/e^3/exp(3*a/b/n)/n^3/((c*(e*x+d)^n)^(3 
/n))-1/2*(e*x+d)*(g*x+f)^2/b/e/n/(a+b*ln(c*(e*x+d)^n))^2+(-d*g+e*f)*(e*x+d 
)*(g*x+f)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))-3/2*(e*x+d)*(g*x+f)^2/b^2/e/n^ 
2/(a+b*ln(c*(e*x+d)^n))
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (e^{\frac {2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-8 e^{\frac {a}{b n}} g (-e f+d g) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+9 g^2 (d+e x)^2 \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-b e e^{\frac {3 a}{b n}} n \left (c (d+e x)^n\right )^{3/n} (f+g x) \left (b e n (f+g x)+a (e f+2 d g+3 e g x)+b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \] Input:

Integrate[(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^3,x]
 

Output:

((d + e*x)*(E^((2*a)/(b*n))*(e*f - d*g)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegra 
lEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 - 8*E^( 
a/(b*n))*g*(-(e*f) + d*g)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[( 
2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])^2 + 9*g^2* 
(d + e*x)^2*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log 
[c*(d + e*x)^n])^2 - b*e*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n)*(f + g*x) 
*(b*e*n*(f + g*x) + a*(e*f + 2*d*g + 3*e*g*x) + b*(2*d*g + e*(f + 3*g*x))* 
Log[c*(d + e*x)^n])))/(2*b^3*e^3*E^((3*a)/(b*n))*n^3*(c*(d + e*x)^n)^(3/n) 
*(a + b*Log[c*(d + e*x)^n])^2)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(753\) vs. \(2(351)=702\).

Time = 3.08 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2847, 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.

\(\displaystyle \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx\)

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2847

\(\displaystyle -\frac {(e f-d g) \left (-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (-\frac {2 (e f-d g) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {3 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2836

\(\displaystyle -\frac {(e f-d g) \left (-\frac {(e f-d g) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )}d(d+e x)}{b e^2 n}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (-\frac {2 (e f-d g) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {3 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2737

\(\displaystyle -\frac {(e f-d g) \left (-\frac {(d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{a+b \log \left (c (d+e x)^n\right )}d\log \left (c (d+e x)^n\right )}{b e^2 n^2}+\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (-\frac {2 (e f-d g) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {3 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle -\frac {(e f-d g) \left (\frac {2 \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (-\frac {2 (e f-d g) \int \frac {f+g x}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {3 \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2846

\(\displaystyle -\frac {(e f-d g) \left (\frac {2 \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b n}-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (\frac {3 \int \left (\frac {(e f-d g)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 g (d+e x) (e f-d g)}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^2 (d+e x)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b n}-\frac {2 (e f-d g) \int \left (\frac {e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b e n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(e f-d g) \left (-\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 \left (\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n}\right )}{b n}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{b e n}+\frac {3 \left (\frac {3 \left (\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^3 n}+\frac {g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}\right )}{b n}-\frac {2 (e f-d g) \left (\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac {g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n}\right )}{b e n}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )}{2 b n}-\frac {(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\)

Input:

Int[(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^3,x]
 

Output:

-1/2*((d + e*x)*(f + g*x)^2)/(b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - ((e*f 
- d*g)*(-(((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*ExpIntegralEi[(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 
*e*n) + (3*((-2*(e*f - d*g)*(((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*L 
og[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*ExpIntegralEi[(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*e*n) + (3*(((e*f - d*g)^2*( 
d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^3*E^(a/(b*n 
))*n*(c*(d + e*x)^n)^n^(-1)) + (2*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[ 
(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^3*E^((2*a)/(b*n))*n*(c*(d + e* 
x)^n)^(2/n)) + (g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n] 
))/(b*n)])/(b*e^3*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n))))/(b*n) - ((d + 
 e*x)*(f + g*x)^2)/(b*e*n*(a + b*Log[c*(d + e*x)^n]))))/(2*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 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 = 3.03 (sec) , antiderivative size = 6505, normalized size of antiderivative = 18.53

method	result	size
risch	\(\text {Expression too large to display}\)	\(6505\)

Input:

int((g*x+f)^2/(a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1090 vs. \(2 (344) = 688\).

Time = 0.12 (sec) , antiderivative size = 1090, normalized size of antiderivative = 3.11 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")
 

Output:

1/2*(8*(a^2*e*f*g - a^2*d*g^2 + (b^2*e*f*g - b^2*d*g^2)*n^2*log(e*x + d)^2 
 + (b^2*e*f*g - b^2*d*g^2)*log(c)^2 + 2*((b^2*e*f*g - b^2*d*g^2)*n*log(c) 
+ (a*b*e*f*g - a*b*d*g^2)*n)*log(e*x + d) + 2*(a*b*e*f*g - a*b*d*g^2)*log( 
c))*e^((b*log(c) + a)/(b*n))*log_integral((e^2*x^2 + 2*d*e*x + d^2)*e^(2*( 
b*log(c) + a)/(b*n))) + (a^2*e^2*f^2 - 2*a^2*d*e*f*g + a^2*d^2*g^2 + (b^2* 
e^2*f^2 - 2*b^2*d*e*f*g + b^2*d^2*g^2)*n^2*log(e*x + d)^2 + (b^2*e^2*f^2 - 
 2*b^2*d*e*f*g + b^2*d^2*g^2)*log(c)^2 + 2*((b^2*e^2*f^2 - 2*b^2*d*e*f*g + 
 b^2*d^2*g^2)*n*log(c) + (a*b*e^2*f^2 - 2*a*b*d*e*f*g + a*b*d^2*g^2)*n)*lo 
g(e*x + d) + 2*(a*b*e^2*f^2 - 2*a*b*d*e*f*g + a*b*d^2*g^2)*log(c))*e^(2*(b 
*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - (b^ 
2*d*e^2*f^2*n^2 + (b^2*e^3*g^2*n^2 + 3*a*b*e^3*g^2*n)*x^3 + ((2*b^2*e^3*f* 
g + b^2*d*e^2*g^2)*n^2 + (4*a*b*e^3*f*g + 5*a*b*d*e^2*g^2)*n)*x^2 + (a*b*d 
*e^2*f^2 + 2*a*b*d^2*e*f*g)*n + ((b^2*e^3*f^2 + 2*b^2*d*e^2*f*g)*n^2 + (a* 
b*e^3*f^2 + 6*a*b*d*e^2*f*g + 2*a*b*d^2*e*g^2)*n)*x + (3*b^2*e^3*g^2*n^2*x 
^3 + (4*b^2*e^3*f*g + 5*b^2*d*e^2*g^2)*n^2*x^2 + (b^2*e^3*f^2 + 6*b^2*d*e^ 
2*f*g + 2*b^2*d^2*e*g^2)*n^2*x + (b^2*d*e^2*f^2 + 2*b^2*d^2*e*f*g)*n^2)*lo 
g(e*x + d) + (3*b^2*e^3*g^2*n*x^3 + (4*b^2*e^3*f*g + 5*b^2*d*e^2*g^2)*n*x^ 
2 + (b^2*e^3*f^2 + 6*b^2*d*e^2*f*g + 2*b^2*d^2*e*g^2)*n*x + (b^2*d*e^2*f^2 
 + 2*b^2*d^2*e*f*g)*n)*log(c))*e^(3*(b*log(c) + a)/(b*n)) + 9*(b^2*g^2*n^2 
*log(e*x + d)^2 + b^2*g^2*log(c)^2 + 2*a*b*g^2*log(c) + a^2*g^2 + 2*(b^...
 

Sympy [F]

\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \] Input:

integrate((g*x+f)**2/(a+b*ln(c*(e*x+d)**n))**3,x)
 

Output:

Integral((f + g*x)**2/(a + b*log(c*(d + e*x)**n))**3, x)
 

Maxima [F]

\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")
 

Output:

-1/2*((3*a*e^2*g^2 + (e^2*g^2*n + 3*e^2*g^2*log(c))*b)*x^3 + ((4*e^2*f*g + 
 5*d*e*g^2)*a + (2*e^2*f*g*n + d*e*g^2*n + (4*e^2*f*g + 5*d*e*g^2)*log(c)) 
*b)*x^2 + (d*e*f^2 + 2*d^2*f*g)*a + (d*e*f^2*n + (d*e*f^2 + 2*d^2*f*g)*log 
(c))*b + ((e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*a + (e^2*f^2*n + 2*d*e*f*g*n + 
 (e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*log(c))*b)*x + (3*b*e^2*g^2*x^3 + (4*e^ 
2*f*g + 5*d*e*g^2)*b*x^2 + (e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*b*x + (d*e*f^ 
2 + 2*d^2*f*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*(9*e^2*g^2*x^ 
2 + e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2 + 2*(4*e^2*f*g + 5*d*e*g^2)*x)/(b^3*e^ 
2*n^2*log((e*x + d)^n) + b^3*e^2*n^2*log(c) + a*b^2*e^2*n^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8422 vs. \(2 (344) = 688\).

Time = 0.24 (sec) , antiderivative size = 8422, normalized size of antiderivative = 23.99 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {too large to display} \] Input:

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

Output:

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