3.1.94 \(\int \frac {(f+g x)^3}{(a+b \log (c (d+e x)^n))^2} \, dx\) [94]

Optimal. Leaf size=339 \[ \frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

(-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))

________________________________________________________________________________________

Rubi [A]
time = 0.57, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \begin {gather*} \frac {9 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} \text {Ei}\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 {6 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} \text {Ei}\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 {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Ei}\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b^2*e^4*E^(a/(b*n))*n^2*(c*(d + e*x
)^n)^n^(-1)) + (6*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^
((2*a)/(b*n))*n^2*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e
*x)^n]))/(b*n)])/(b^2*e^4*E^((3*a)/(b*n))*n^2*(c*(d + e*x)^n)^(3/n)) + (4*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a
+ b*Log[c*(d + e*x)^n]))/(b*n)])/(b^2*e^4*E^((4*a)/(b*n))*n^2*(c*(d + e*x)^n)^(4/n)) - ((d + e*x)*(f + g*x)^3)
/(b*e*n*(a + b*Log[c*(d + e*x)^n]))

Rule 2209

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

Rule 2337

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[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 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(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 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[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 2437

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

Rule 2446

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(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 2447

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] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[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] && LtQ[p, -1] && GtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {4 \int \frac {(f+g x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b n}-\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 {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\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 (e f-d g)^2 (d+e x)}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {3 g^2 (e f-d g) (d+e x)^2}{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 (e f-d g) (d+e x)}{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 )}+\frac {\left (4 g^3\right ) \int \frac {(d+e x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac {\left (3 g^2 (e f-d g)\right ) \int \frac {(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac {\left (12 g^2 (e f-d g)\right ) \int \frac {(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac {\left (6 g (e f-d g)^2\right ) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac {\left (12 g (e f-d g)^2\right ) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}-\frac {\left (3 (e f-d g)^3\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}+\frac {\left (4 (e f-d g)^3\right ) \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b e^3 n}\\ &=-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (4 g^3\right ) \text {Subst}\left (\int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac {\left (3 g^2 (e f-d g)\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac {\left (12 g^2 (e f-d g)\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac {\left (6 g (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac {\left (12 g (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}-\frac {\left (3 (e f-d g)^3\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}+\frac {\left (4 (e f-d g)^3\right ) \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e^4 n}\\ &=-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left (4 g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\frac {\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}+\frac {\left (12 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\frac {\left (6 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}+\frac {\left (12 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}-\frac {\left (3 (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}+\frac {\left (4 (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e^4 n^2}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\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} \text {Ei}\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} \text {Ei}\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} \text {Ei}\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 )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1674\) vs. \(2(339)=678\).
time = 0.62, size = 1674, normalized size = 4.94 \begin {gather*} \frac {e^{-\frac {4 a}{b n}} \left (c (d+e x)^n\right )^{-4/n} \left (-b d e^3 e^{\frac {4 a}{b n}} f^3 n \left (c (d+e x)^n\right )^{4/n}-b e^4 e^{\frac {4 a}{b n}} f^3 n x \left (c (d+e x)^n\right )^{4/n}-3 b d e^3 e^{\frac {4 a}{b n}} f^2 g n x \left (c (d+e x)^n\right )^{4/n}-3 b e^4 e^{\frac {4 a}{b n}} f^2 g n x^2 \left (c (d+e x)^n\right )^{4/n}-3 b d e^3 e^{\frac {4 a}{b n}} f g^2 n x^2 \left (c (d+e x)^n\right )^{4/n}-3 b e^4 e^{\frac {4 a}{b n}} f g^2 n x^3 \left (c (d+e x)^n\right )^{4/n}-b d e^3 e^{\frac {4 a}{b n}} g^3 n x^3 \left (c (d+e x)^n\right )^{4/n}-b e^4 e^{\frac {4 a}{b n}} g^3 n x^4 \left (c (d+e x)^n\right )^{4/n}+a e^3 e^{\frac {3 a}{b n}} f^3 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )-3 a d e^2 e^{\frac {3 a}{b n}} f^2 g (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+3 a d^2 e e^{\frac {3 a}{b n}} f g^2 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )-a d^3 e^{\frac {3 a}{b n}} g^3 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+6 a e^2 e^{\frac {2 a}{b n}} f^2 g (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-12 a d e e^{\frac {2 a}{b n}} f g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+6 a d^2 e^{\frac {2 a}{b n}} g^3 (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+9 a e e^{\frac {a}{b n}} f g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-9 a d e^{\frac {a}{b n}} g^3 (d+e x)^3 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+4 a g^3 (d+e x)^4 \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+b e^3 e^{\frac {3 a}{b n}} f^3 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-3 b d e^2 e^{\frac {3 a}{b n}} f^2 g (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+3 b d^2 e e^{\frac {3 a}{b n}} f g^2 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-b d^3 e^{\frac {3 a}{b n}} g^3 (d+e x) \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+6 b e^2 e^{\frac {2 a}{b n}} f^2 g (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-12 b d e e^{\frac {2 a}{b n}} f g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+6 b d^2 e^{\frac {2 a}{b n}} g^3 (d+e x)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+9 b e e^{\frac {a}{b n}} f g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )-9 b d e^{\frac {a}{b n}} g^3 (d+e x)^3 \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )+4 b g^3 (d+e x)^4 \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \log \left (c (d+e x)^n\right )\right )}{b^2 e^4 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(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)*ExpIn
tegralEi[(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)*ExpIntegralEi[(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]))/(b*n)] + 4*a*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(
d + e*x)^n]))/(b*n)] + b*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)]*Log[c*(d + e*x)^n] - 3*b*d*e^2*E^((3*a)/(b*n))*f^2*g*(d + e*x)*(c*(d + e*x)^n)^(3/n)*ExpInt
egralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] + 3*b*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)]*Log[c*(d + e*x)^n] - b*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)]*Log[c*(d + e*x)^n] + 6*b*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)]*L
og[c*(d + e*x)^n] - 12*b*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*L
og[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] + 6*b*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)]*Log[c*(d + e*x)^n] + 9*b*e*E^(a/(b*n))*f*g^2*(d + e*x)^3*(
c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] - 9*b*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]))/(b*n)]*Log[c*(d + e*x)^
n] + 4*b*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n])/(b^2*e^4*E^((
4*a)/(b*n))*n^2*(c*(d + e*x)^n)^(4/n)*(a + b*Log[c*(d + e*x)^n]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.23, size = 9517, normalized size = 28.07

method	result	size
risch	\(\text {Expression too large to display}\)	\(9517\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 676, normalized size = 1.99 \begin {gather*} -\frac {{\left (9 \, {\left (a d g^{3} - a f g^{2} e + {\left (b d g^{3} n - b f g^{2} n e\right )} \log \left (x e + d\right ) + {\left (b d g^{3} - b f g^{2} e\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) - 6 \, {\left (a d^{2} g^{3} - 2 \, a d f g^{2} e + a f^{2} g e^{2} + {\left (b d^{2} g^{3} n - 2 \, b d f g^{2} n e + b f^{2} g n e^{2}\right )} \log \left (x e + d\right ) + {\left (b d^{2} g^{3} - 2 \, b d f g^{2} e + b f^{2} g e^{2}\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 (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + {\left (a d^{3} g^{3} - 3 \, a d^{2} f g^{2} e + 3 \, a d f^{2} g e^{2} - a f^{3} e^{3} + {\left (b d^{3} g^{3} n - 3 \, b d^{2} f g^{2} n e + 3 \, b d f^{2} g n e^{2} - b f^{3} n e^{3}\right )} \log \left (x e + d\right ) + {\left (b d^{3} g^{3} - 3 \, b d^{2} f g^{2} e + 3 \, b d f^{2} g e^{2} - b f^{3} e^{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 (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) + {\left ({\left (b g^{3} n x^{4} + 3 \, b f g^{2} n x^{3} + 3 \, b f^{2} g n x^{2} + b f^{3} n x\right )} e^{4} + {\left (b d g^{3} n x^{3} + 3 \, b d f g^{2} n x^{2} + 3 \, b d f^{2} g n x + b d f^{3} n\right )} e^{3}\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \, {\left (b g^{3} n \log \left (x e + d\right ) + b g^{3} \log \left (c\right ) + a g^{3}\right )} \operatorname {log\_integral}\left ({\left (x^{4} e^{4} + 4 \, d x^{3} e^{3} + 6 \, d^{2} x^{2} e^{2} + 4 \, d^{3} x e + 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} n^{3} e^{4} \log \left (x e + d\right ) + b^{3} n^{2} e^{4} \log \left (c\right ) + a b^{2} n^{2} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{3}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3475 vs. \(2 (354) = 708\).
time = 5.70, size = 3475, normalized size = 10.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e + d)^4*b*g^3*n*e^6/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*(x*e + d)^3*b*
d*g^3*n*e^6/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - 3*(x*e + d)^2*b*d^2*g^3*n*e^6
/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + (x*e + d)*b*d^3*g^3*n*e^6/(b^3*n^3*e^10*
log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - b*d^3*g^3*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a
/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 3*(x*e
+ d)^3*b*f*g^2*n*e^7/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 6*(x*e + d)^2*b*d*f
*g^2*n*e^7/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - 3*(x*e + d)*b*d^2*f*g^2*n*e^7/
(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*b*d^2*f*g^2*n*Ei(log(c)/n + a/(b*n) + l
og(x*e + d))*e^(-a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)
*c^(1/n)) + 6*b*d^2*g^3*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 6)*log(x*e + d)/((b^3*n^
3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - b*d^3*g^3*Ei(log(c)/n + a/(b*n) + log(x
*e + d))*e^(-a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n))
- 3*(x*e + d)^2*b*f^2*g*n*e^8/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*(x*e + d)
*b*d*f^2*g*n*e^8/(b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10) - a*d^3*g^3*Ei(log(c)/n +
a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c
^(1/n)) - 3*b*d*f^2*g*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)*log(x*e + d)/((b^3*n^3*e^10*log
(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*b*d*f*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(
x*e + d))*e^(-2*a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*
c^(2/n)) - 9*b*d*g^3*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e
^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 3*b*d^2*f*g^2*Ei(log(c)/n + a/(b*n) + log(
x*e + d))*e^(-a/(b*n) + 7)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n))
+ 6*b*d^2*g^3*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e +
d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - (x*e + d)*b*f^3*n*e^9/(b^3*n^3*e^10*log(x*e + d) + b^3*n
^2*e^10*log(c) + a*b^2*n^2*e^10) + 3*a*d^2*f*g^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 7)/((b^3*
n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*a*d^2*g^3*Ei(2*log(c)/n + 2*a/(b*n)
+ 2*log(x*e + d))*e^(-2*a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2
/n)) + b*f^3*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 9)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d)
+ b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*b*f^2*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^
(-2*a/(b*n) + 8)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) + 9
*b*f*g^2*n*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 7)*log(x*e + d)/((b^3*n^3*e^10*log(x*e
+ d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 4*b*g^3*n*Ei(4*log(c)/n + 4*a/(b*n) + 4*log(x*e + d))*
e^(-4*a/(b*n) + 6)*log(x*e + d)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(4/n)) -
3*b*d*f^2*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n
^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*b*d*f*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(
b*n) + 7)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/n)) - 9*b*d*g^3*Ei(3
*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*l
og(c) + a*b^2*n^2*e^10)*c^(3/n)) - 3*a*d*f^2*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 8)/((b^3*n^
3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) - 12*a*d*f*g^2*Ei(2*log(c)/n + 2*a/(b*n)
+ 2*log(x*e + d))*e^(-2*a/(b*n) + 7)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(2/
n)) - 9*a*d*g^3*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 6)/((b^3*n^3*e^10*log(x*e + d) + b
^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + b*f^3*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 9)*l
og(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(1/n)) + 6*b*f^2*g*Ei(2*log(c)/n +
2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 8)*log(c)/((b^3*n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b
^2*n^2*e^10)*c^(2/n)) + 9*b*f*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 7)*log(c)/((b^3*
n^3*e^10*log(x*e + d) + b^3*n^2*e^10*log(c) + a*b^2*n^2*e^10)*c^(3/n)) + 4*b*g^3*Ei(4*log(c)/n + 4*a/(b*n) + 4
*log(x*e + d))*e^(-4*a/(b*n) + 6)*log(c)/((b^3*...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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