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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347, 2334} \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e^2 n^3}+\frac {2 g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^3 e^2 n^3}+\frac {(d+e x) (e f-d g)}{2 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {(d+e x) (f+g x)}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[In]

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

[Out]

((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(2*b^3*e^2*E^(a/(b*n))*n^3*(c*(d + e*x
)^n)^n^(-1)) + (2*g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b^3*e^2*E^((2*a)/(b*n))*
n^3*(c*(d + e*x)^n)^(2/n)) - ((d + e*x)*(f + g*x))/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) + ((e*f - d*g)*(d +
e*x))/(2*b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - ((d + e*x)*(f + g*x))/(b^2*e*n^2*(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 2334

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

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.98 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=-\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-4 g (d+e x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b e^{\frac {2 a}{b n}} n \left (c (d+e x)^n\right )^{2/n} \left (b e n (f+g x)+a (e f+d g+2 e g x)+b (d g+e (f+2 g x)) \log \left (c (d+e x)^n\right )\right )\right )}{2 b^3 e^2 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.25 (sec) , antiderivative size = 3114, normalized size of antiderivative = 11.93

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

[In]

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

[Out]

-(2*ln(c)*b*d^2*g-I*Pi*b*d^2*g*csgn(I*c*(e*x+d)^n)^3+6*b*d*e*g*x*ln((e*x+d)^n)-3*I*Pi*b*d*e*g*x*csgn(I*c)*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+6*ln(c)*b*d*e*g*x+2*b*e^2*g*n*x^2+2*b*e^2*f*n*x+2*b*e^2*f*x*ln((e*x+d)^n)+2*
b*d*e*f*ln((e*x+d)^n)+I*Pi*b*d^2*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*d^2*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)^2+2*ln(c)*b*d*e*f-2*I*Pi*b*e^2*g*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3*I*Pi*b*d*e*g*x*
csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+2*b*d*e*g*n*x+4*b*e^2*g*x^2*ln((e*x+d)^n)+4*ln(c)*b*e^2*g*x^2+2*ln(c)*b*e^2*f*
x+6*a*d*e*g*x+3*I*Pi*b*d*e*g*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*e^2*f*x*csgn(I*c*(e*x+d)^n)^3-2*
I*Pi*b*e^2*g*x^2*csgn(I*c*(e*x+d)^n)^3+2*a*d^2*g+4*a*e^2*g*x^2+2*a*e^2*f*x+2*b*d^2*g*ln((e*x+d)^n)-I*Pi*b*d*e*
f*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*I*Pi*b*e^2*g*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*d^
2*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*I*Pi*b*e^2*g*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
-3*I*Pi*b*d*e*g*x*csgn(I*c*(e*x+d)^n)^3-I*Pi*b*d*e*f*csgn(I*c*(e*x+d)^n)^3+2*b*d*e*f*n-I*Pi*b*e^2*f*x*csgn(I*c
)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*a*d*e*f+I*Pi*b*d*e*f*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*
e^2*f*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*d*e*f*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*b*e^2*f*x*csgn(I*(e*
x+d)^n)*csgn(I*c*(e*x+d)^n)^2)/b^2/e^2/n^2/(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(
I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*l
n((e*x+d)^n)+2*b*ln(c)+2*a)^2-1/2/b^3/n^3*f*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*
csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^
2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(
e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*
(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*x-1/2/e/b^3/n^3*f*c^(-1/n)*((e*x+d)^n)^(-1/
n)*exp(-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*
Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*
Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d-
2/b^3/n^3*g*c^(-2/n)*((e*x+d)^n)^(-2/n)*exp(-(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csg
n(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a
)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x
+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+
d)^n)-n*ln(e*x+d))+2*a)/b/n)*x^2-4/e/b^3/n^3*g*c^(-2/n)*((e*x+d)^n)^(-2/n)*exp(-(-I*b*Pi*csgn(I*c*(e*x+d)^n)*c
sgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x
+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*
x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d*x-2/e^2/b^3/n^3*g*c^(-2/n)*((e*x+d)^n)^(-2/n
)*exp(-(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*cs
gn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-2*ln(e*x+d)-(-I*b*Pi*csgn
(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*d^2+1/2/e
/b^3/n^3*d*g*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*P
i*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*
b+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I
*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(l
n((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)*x+1/2/e^2/b^3/n^3*d^2*g*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(-I*b*Pi*csgn
(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*a)/b/n)*Ei(1,-ln(e*x+d)-1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*cs
gn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*
b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/b/n)

Fricas [B] (verification not implemented)

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

Time = 0.32 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.25 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\frac {{\left ({\left ({\left (b^{2} e f - b^{2} d g\right )} n^{2} \log \left (e x + d\right )^{2} + a^{2} e f - a^{2} d g + {\left (b^{2} e f - b^{2} d g\right )} \log \left (c\right )^{2} + 2 \, {\left ({\left (b^{2} e f - b^{2} d g\right )} n \log \left (c\right ) + {\left (a b e f - a b d g\right )} n\right )} \log \left (e x + d\right ) + 2 \, {\left (a b e f - a b d g\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) - {\left (b^{2} d e f n^{2} + {\left (b^{2} e^{2} g n^{2} + 2 \, a b e^{2} g n\right )} x^{2} + {\left (a b d e f + a b d^{2} g\right )} n + {\left ({\left (b^{2} e^{2} f + b^{2} d e g\right )} n^{2} + {\left (a b e^{2} f + 3 \, a b d e g\right )} n\right )} x + {\left (2 \, b^{2} e^{2} g n^{2} x^{2} + {\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n^{2} x + {\left (b^{2} d e f + b^{2} d^{2} g\right )} n^{2}\right )} \log \left (e x + d\right ) + {\left (2 \, b^{2} e^{2} g n x^{2} + {\left (b^{2} e^{2} f + 3 \, b^{2} d e g\right )} n x + {\left (b^{2} d e f + b^{2} d^{2} g\right )} n\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 4 \, {\left (b^{2} g n^{2} \log \left (e x + d\right )^{2} + b^{2} g \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) + a^{2} g + 2 \, {\left (b^{2} g n \log \left (c\right ) + a b g n\right )} \log \left (e x + d\right )\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{2 \, {\left (b^{5} e^{2} n^{5} \log \left (e x + d\right )^{2} + b^{5} e^{2} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e^{2} n^{3} \log \left (c\right ) + a^{2} b^{3} e^{2} n^{3} + 2 \, {\left (b^{5} e^{2} n^{4} \log \left (c\right ) + a b^{4} e^{2} n^{4}\right )} \log \left (e x + d\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

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

Time = 0.43 (sec) , antiderivative size = 4112, normalized size of antiderivative = 15.75 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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