∫ (f+gx)2 _______________________________

3.100 $\int \frac{(f+g x)^2}{(a+b \log (c (d+e x)^n))^3} \, dx$

Optimal. Leaf size=351 \[ \frac{4 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac{9 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac{(d+e x) (f+g x) (e f-d g)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[Out]

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

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

((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*

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

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

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

________________________________________________________________________________________

Rubi [A] time = 0.864587, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 7, integrand size = 24, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.292, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{4 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^3 n^3}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac{9 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{2 b^3 e^3 n^3}+\frac{(d+e x) (f+g x) (e f-d g)}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{3 (d+e x) (f+g x)^2}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{(d+e x) (f+g x)^2}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

((2*a)/(b*n))*n^3*(c*(d + e*x)^n)^(2/n)) + (9*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*

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

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

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

Rule 2400

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]

Rule 2399

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 2389

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 2300

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 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral

Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True

Rule 2390

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}

, x]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=-\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{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{(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{(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 )}+\frac{9 \int \frac{(f+g x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 n^2}-\frac{(2 (e f-d g)) \int \frac{f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}-\frac{(3 (e f-d g)) \int \frac{f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e n^2}+\frac{(e f-d g)^2 \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}\\ &=-\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 )}+\frac{9 \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}{2 b^2 n^2}-\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^2 e n^2}-\frac{(3 (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^2 e n^2}+\frac{(e f-d g)^2 \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}\\ &=-\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 )}+\frac{\left (9 g^2\right ) \int \frac{(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}-\frac{(2 g (e f-d g)) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac{(3 g (e f-d g)) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac{(9 g (e f-d g)) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac{\left (2 (e f-d g)^2\right ) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}-\frac{\left (3 (e f-d g)^2\right ) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{b^2 e^2 n^2}+\frac{\left (9 (e f-d g)^2\right ) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{2 b^2 e^2 n^2}+\frac{\left ((e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^2 (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^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 )}+\frac{\left (9 g^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2}-\frac{(2 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{(3 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{(9 g (e f-d g)) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{\left (2 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}-\frac{\left (3 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b^2 e^3 n^2}+\frac{\left (9 (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e^3 n^2}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^2 (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^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 )}+\frac{\left (9 g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3}-\frac{\left (2 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (3 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (9 g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (2 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}-\frac{\left (3 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b^2 e^3 n^3}+\frac{\left (9 (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e^3 n^3}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^2 (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 )}{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} \text{Ei}\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} \text{Ei}\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 )}\\ \end{align*}

Mathematica [A] time = 1.48522, size = 351, normalized size = 1. \[ \frac{e^{-\frac{3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (e^{\frac{2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )-8 g e^{\frac{a}{b n}} (d+e x) (d g-e f) \left (c (d+e x)^n\right )^{\frac{1}{n}} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+9 g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-b e n e^{\frac{3 a}{b n}} (f+g x) \left (c (d+e x)^n\right )^{3/n} \left (a (2 d g+e f+3 e g x)+b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )+b e n (f+g x)\right )\right )}{2 b^3 e^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F] time = 3.762, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{2}}{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, a e^{2} g^{2} +{\left (e^{2} g^{2} n + 3 \, e^{2} g^{2} \log \left (c\right )\right )} b\right )} x^{3} +{\left ({\left (4 \, e^{2} f g + 5 \, d e g^{2}\right )} a +{\left (2 \, e^{2} f g n + d e g^{2} n +{\left (4 \, e^{2} f g + 5 \, d e g^{2}\right )} \log \left (c\right )\right )} b\right )} x^{2} +{\left (d e f^{2} + 2 \, d^{2} f g\right )} a +{\left (d e f^{2} n +{\left (d e f^{2} + 2 \, d^{2} f g\right )} \log \left (c\right )\right )} b +{\left ({\left (e^{2} f^{2} + 6 \, d e f g + 2 \, d^{2} g^{2}\right )} a +{\left (e^{2} f^{2} n + 2 \, d e f g n +{\left (e^{2} f^{2} + 6 \, d e f g + 2 \, d^{2} g^{2}\right )} \log \left (c\right )\right )} b\right )} x +{\left (3 \, b e^{2} g^{2} x^{3} +{\left (4 \, e^{2} f g + 5 \, d e g^{2}\right )} b x^{2} +{\left (e^{2} f^{2} + 6 \, d e f g + 2 \, d^{2} g^{2}\right )} b x +{\left (d e f^{2} + 2 \, d^{2} f g\right )} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \,{\left (b^{4} e^{2} n^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{4} e^{2} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} e^{2} n^{2} \log \left (c\right ) + a^{2} b^{2} e^{2} n^{2} + 2 \,{\left (b^{4} e^{2} n^{2} \log \left (c\right ) + a b^{3} e^{2} n^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac{9 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + 2 \, d^{2} g^{2} + 2 \,{\left (4 \, e^{2} f g + 5 \, d e g^{2}\right )} x}{2 \,{\left (b^{3} e^{2} n^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b^{3} e^{2} n^{2} \log \left (c\right ) + a b^{2} e^{2} n^{2}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*l

og(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)

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Fricas [B] time = 2.36863, size = 2404, normalized size = 6.85 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*lo

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

x + d)^2 + 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*(b^5*e^3*n^4*log(c) + a*b^4*e^3

*n^4)*log(e*x + d))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] time = 2.04016, size = 11335, normalized size = 32.29 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 2*(x*e + d

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

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

*g^2*n^2*e^3*log(x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*lo

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

n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 3)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x

*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^

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

+ 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + (x*e + d)^

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

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

^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log

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

g(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b

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

2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(

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

b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*lo

g(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) - 4*b^2*d*g^2*n^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*lo

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

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

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

*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 2*(x*e + d)^2*b^2*d

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

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

/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*

log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + b^2*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(

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

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

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

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

*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*

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

4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2

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

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

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

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

x*e + d)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*

n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + a*b*d^2*g^2*n*Ei(log(c)/n + a/(b*n) + log(x*e +

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

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

*n^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 5)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^

5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) +

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

x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b

^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n)) + 9/2*b^2*g^2*n^2*Ei(3*log(c)/n + 3*a

/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 3)*log(x*e + d)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*

e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6

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

) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + (x*e + d

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

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

b*n) + log(x*e + d))*e^(-a/(b*n) + 4)*log(x*e + d)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e

+ d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)

*c^(1/n)) - 8*b^2*d*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 3)*log(x*e + d)*log(c)/(

(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*l

og(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n)) + 1/2*b^2*d^2*g^2*Ei(log(c)/n + a/(b*n) + log(x*e

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

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

*b^2*f^2*n^2*e^5/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d

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

6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2

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

^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^

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

og(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*

b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) - 8*a*b*d*g^2*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(

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

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

*f^2*n*e^5*log(c)/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e +

d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + a*b*d^2*g^2*Ei(log(c)/n + a/(b*n) + lo

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

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

Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 5)*log(x*e + d)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^

5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) +

a^2*b^3*n^3*e^6)*c^(1/n)) + 8*b^2*f*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)*log(x*

e + d)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d)

+ b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n)) + 9*b^2*g^2*n*Ei(3*log(c)/n + 3*a/

(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 3)*log(x*e + d)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*lo

g(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3

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

*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*

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

b*n) + 3)*log(c)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e

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

/(b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*

log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6) + 1/2*a^2*d^2*g^2*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e

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

) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) + a*b*f^2*n*Ei(log(c)/n + a/(b*n

) + log(x*e + d))*e^(-a/(b*n) + 5)*log(x*e + d)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(

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

+ 8*a*b*f*g*n*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)*log(x*e + d)/((b^5*n^5*e^6*log(x*

e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n

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

*n) + 3)*log(x*e + d)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x

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

+ a/(b*n) + log(x*e + d))*e^(-a/(b*n) + 4)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*lo

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

) - 8*a*b*d*g^2*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 3)*log(c)/((b^5*n^5*e^6*log(x*e +

d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e

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

(c)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^

3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) + 4*b^2*f*g*Ei(2*log(c)/n + 2*a/(b*n) + 2*

log(x*e + d))*e^(-2*a/(b*n) + 4)*log(c)^2/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2

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

b^2*g^2*Ei(3*log(c)/n + 3*a/(b*n) + 3*log(x*e + d))*e^(-3*a/(b*n) + 3)*log(c)^2/((b^5*n^5*e^6*log(x*e + d)^2 +

2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log

^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 +

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

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

d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n)) + a*b*f^2*Ei(log(c)/n + a/(b*n

) + log(x*e + d))*e^(-a/(b*n) + 5)*log(c)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2

*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(1/n)) + 8*a*

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

^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c)

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

((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*

log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(3/n)) + 1/2*a^2*f^2*Ei(log(c)/n + a/(b*n) + log(x*e +

d))*e^(-a/(b*n) + 5)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*

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

2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) + 4)/((b^5*n^5*e^6*log(x*e + d)^2 + 2*b^5*n^4*e^6*log(x*e + d)*log(

c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c) + a^2*b^3*n^3*e^6)*c^(2/n))

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

b^5*n^4*e^6*log(x*e + d)*log(c) + 2*a*b^4*n^4*e^6*log(x*e + d) + b^5*n^3*e^6*log(c)^2 + 2*a*b^4*n^3*e^6*log(c)

+ a^2*b^3*n^3*e^6)*c^(3/n))

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