Integrand size = 24, antiderivative size = 339 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
(-d*g+e*f)^3*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(a/b/n)/n^2/ ((c*(e*x+d)^n)^(1/n))+6*g*(-d*g+e*f)^2*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n) )/b/n)/b^2/e^4/exp(2*a/b/n)/n^2/((c*(e*x+d)^n)^(2/n))+9*g^2*(-d*g+e*f)*(e* x+d)^3*Ei(3*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(3*a/b/n)/n^2/((c*(e*x+d )^n)^(3/n))+4*g^3*(e*x+d)^4*Ei(4*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^4/exp(4* a/b/n)/n^2/((c*(e*x+d)^n)^(4/n))-(e*x+d)*(g*x+f)^3/b/e/n/(a+b*ln(c*(e*x+d) ^n))
Leaf count is larger than twice the leaf count of optimal. \(1674\) vs. \(2(339)=678\).
Time = 0.46 (sec) , antiderivative size = 1674, normalized size of antiderivative = 4.94 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx =\text {Too large to display} \]
(-(b*d*e^3*E^((4*a)/(b*n))*f^3*n*(c*(d + e*x)^n)^(4/n)) - b*e^4*E^((4*a)/( b*n))*f^3*n*x*(c*(d + e*x)^n)^(4/n) - 3*b*d*e^3*E^((4*a)/(b*n))*f^2*g*n*x* (c*(d + e*x)^n)^(4/n) - 3*b*e^4*E^((4*a)/(b*n))*f^2*g*n*x^2*(c*(d + e*x)^n )^(4/n) - 3*b*d*e^3*E^((4*a)/(b*n))*f*g^2*n*x^2*(c*(d + e*x)^n)^(4/n) - 3* b*e^4*E^((4*a)/(b*n))*f*g^2*n*x^3*(c*(d + e*x)^n)^(4/n) - b*d*e^3*E^((4*a) /(b*n))*g^3*n*x^3*(c*(d + e*x)^n)^(4/n) - b*e^4*E^((4*a)/(b*n))*g^3*n*x^4* (c*(d + e*x)^n)^(4/n) + a*e^3*E^((3*a)/(b*n))*f^3*(d + e*x)*(c*(d + e*x)^n )^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)] - 3*a*d*e^2*E^((3* a)/(b*n))*f^2*g*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c *(d + e*x)^n])/(b*n)] + 3*a*d^2*e*E^((3*a)/(b*n))*f*g^2*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)] - a*d^3*E^(( 3*a)/(b*n))*g^3*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpIntegralEi[(a + b*Log[c *(d + e*x)^n])/(b*n)] + 6*a*e^2*E^((2*a)/(b*n))*f^2*g*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 12*a*d *e*E^((2*a)/(b*n))*f*g^2*(d + e*x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[( 2*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + 6*a*d^2*E^((2*a)/(b*n))*g^3*(d + e* x)^2*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b *n)] + 9*a*e*E^(a/(b*n))*f*g^2*(d + e*x)^3*(c*(d + e*x)^n)^n^(-1)*ExpInteg ralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)] - 9*a*d*E^(a/(b*n))*g^3*(d + e *x)^3*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]...
Time = 1.19 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.73, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2847, 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.
| \(\displaystyle \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {3 (e f-d g) \int \frac {(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )}dx}{b e n}+\frac {4 \int \frac {(f+g x)^3}{a+b \log \left (c (d+e x)^n\right )}dx}{b n}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\) |
| \(\Big \downarrow \) 2846 |
| \(\displaystyle \frac {4 \int \left (\frac {(e f-d g)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {3 g (d+e x) (e f-d g)^2}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {3 g^2 (d+e x)^2 (e f-d g)}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^3 (d+e x)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b n}-\frac {3 (e f-d g) \int \left (\frac {(e f-d g)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {2 g (d+e x) (e f-d g)}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^2 (d+e x)^2}{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right )dx}{b e n}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {3 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {3 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac {g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}\right )}{b n}-\frac {3 (e f-d g) \left (\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^3 n}+\frac {g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^3 n}\right )}{b e n}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}\) |
(-3*(e*f - d*g)*(((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*ExpIntegr alEi[(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*e*n) + (4*(((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[( a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^4*E^(a/(b*n))*n*(c*(d + e*x)^n)^n^( -1)) + (3*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e *x)^n]))/(b*n)])/(b*e^4*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)) + (3*g^2* (e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n) ])/(b*e^4*E^((3*a)/(b*n))*n*(c*(d + e*x)^n)^(3/n)) + (g^3*(d + e*x)^4*ExpI ntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^4*E^((4*a)/(b*n))*n* (c*(d + e*x)^n)^(4/n))))/(b*n) - ((d + e*x)*(f + g*x)^3)/(b*e*n*(a + b*Log [c*(d + e*x)^n]))
3.1.94.3.1 Defintions of rubi rules used
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.70 (sec) , antiderivative size = 9517, normalized size of antiderivative = 28.07
Time = 0.32 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.01 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {{\left (9 \, {\left (a e f g^{2} - a d g^{3} + {\left (b e f g^{2} - b d g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e f g^{2} - b d g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 6 \, {\left (a e^{2} f^{2} g - 2 \, a d e f g^{2} + a d^{2} g^{3} + {\left (b e^{2} f^{2} g - 2 \, b d e f g^{2} + b d^{2} g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e^{2} f^{2} g - 2 \, b d e f g^{2} + b d^{2} g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\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 ) + {\left (a e^{3} f^{3} - 3 \, a d e^{2} f^{2} g + 3 \, a d^{2} e f g^{2} - a d^{3} g^{3} + {\left (b e^{3} f^{3} - 3 \, b d e^{2} f^{2} g + 3 \, b d^{2} e f g^{2} - b d^{3} g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e^{3} f^{3} - 3 \, b d e^{2} f^{2} g + 3 \, b d^{2} e f g^{2} - b d^{3} g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{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 e^{4} g^{3} n x^{4} + b d e^{3} f^{3} n + {\left (3 \, b e^{4} f g^{2} + b d e^{3} g^{3}\right )} n x^{3} + 3 \, {\left (b e^{4} f^{2} g + b d e^{3} f g^{2}\right )} n x^{2} + {\left (b e^{4} f^{3} + 3 \, b d e^{3} f^{2} g\right )} n x\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 4 \, {\left (b g^{3} n \log \left (e x + d\right ) + b g^{3} \log \left (c\right ) + a g^{3}\right )} \operatorname {log\_integral}\left ({\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} e^{4} n^{3} \log \left (e x + d\right ) + b^{3} e^{4} n^{2} \log \left (c\right ) + a b^{2} e^{4} n^{2}} \]
(9*(a*e*f*g^2 - a*d*g^3 + (b*e*f*g^2 - b*d*g^3)*n*log(e*x + d) + (b*e*f*g^ 2 - b*d*g^3)*log(c))*e^((b*log(c) + a)/(b*n))*log_integral((e^3*x^3 + 3*d* e^2*x^2 + 3*d^2*e*x + d^3)*e^(3*(b*log(c) + a)/(b*n))) + 6*(a*e^2*f^2*g - 2*a*d*e*f*g^2 + a*d^2*g^3 + (b*e^2*f^2*g - 2*b*d*e*f*g^2 + b*d^2*g^3)*n*lo g(e*x + d) + (b*e^2*f^2*g - 2*b*d*e*f*g^2 + b*d^2*g^3)*log(c))*e^(2*(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*e^3*f^3 - 3*a*d*e^2*f^2*g + 3*a*d^2*e*f*g^2 - a*d^3*g^3 + (b *e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b*d^2*e*f*g^2 - b*d^3*g^3)*n*log(e*x + d) + (b*e^3*f^3 - 3*b*d*e^2*f^2*g + 3*b*d^2*e*f*g^2 - b*d^3*g^3)*log(c))*e^(3* (b*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) - ( b*e^4*g^3*n*x^4 + b*d*e^3*f^3*n + (3*b*e^4*f*g^2 + b*d*e^3*g^3)*n*x^3 + 3* (b*e^4*f^2*g + b*d*e^3*f*g^2)*n*x^2 + (b*e^4*f^3 + 3*b*d*e^3*f^2*g)*n*x)*e ^(4*(b*log(c) + a)/(b*n)) + 4*(b*g^3*n*log(e*x + d) + b*g^3*log(c) + a*g^3 )*log_integral((e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4)*e ^(4*(b*log(c) + a)/(b*n))))*e^(-4*(b*log(c) + a)/(b*n))/(b^3*e^4*n^3*log(e *x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)
\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {\left (f + g x\right )^{3}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {{\left (g x + f\right )}^{3}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]
-(e*g^3*x^4 + d*f^3 + (3*e*f*g^2 + d*g^3)*x^3 + 3*(e*f^2*g + d*f*g^2)*x^2 + (e*f^3 + 3*d*f^2*g)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) + a*b* e*n) + integrate((4*e*g^3*x^3 + e*f^3 + 3*d*f^2*g + 3*(3*e*f*g^2 + d*g^3)* x^2 + 6*(e*f^2*g + d*f*g^2)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) + a*b*e*n), x)
Leaf count of result is larger than twice the leaf count of optimal. 3473 vs. \(2 (341) = 682\).
Time = 0.42 (sec) , antiderivative size = 3473, normalized size of antiderivative = 10.24 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Too large to display} \]
b*e^3*f^3*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*log(e*x + d )/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)*c^(1/n) ) - 3*b*d*e^2*f^2*g*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(-a/(b*n))*l og(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^ 2)*c^(1/n)) + 3*b*d^2*e*f*g^2*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))*e^(- a/(b*n))*log(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a* b^2*e^4*n^2)*c^(1/n)) - b*d^3*g^3*n*Ei(log(c)/n + a/(b*n) + log(e*x + d))* e^(-a/(b*n))*log(e*x + d)/((b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2)*c^(1/n)) - (e*x + d)*b*e^3*f^3*n/(b^3*e^4*n^3*log(e*x + d ) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - 3*(e*x + d)^2*b*e^2*f^2*g*n/(b^3 *e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) + 3*(e*x + d)* b*d*e^2*f^2*g*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4 *n^2) - 3*(e*x + d)^3*b*e*f*g^2*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2* log(c) + a*b^2*e^4*n^2) + 6*(e*x + d)^2*b*d*e*f*g^2*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - 3*(e*x + d)*b*d^2*e*f*g^2*n/ (b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - (e*x + d )^4*b*g^3*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2 ) + 3*(e*x + d)^3*b*d*g^3*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) - 3*(e*x + d)^2*b*d^2*g^3*n/(b^3*e^4*n^3*log(e*x + d) + b^3*e^4*n^2*log(c) + a*b^2*e^4*n^2) + (e*x + d)*b*d^3*g^3*n/(b^3*e^4*n^...
Timed out. \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {{\left (f+g\,x\right )}^3}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]