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

Optimal result

Integrand size = 24, antiderivative size = 339 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {6 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {9 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[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] (verified)

Time = 0.53 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {9 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {6 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^4 n^2}+\frac {4 g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \operatorname {ExpIntegralEi}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^4 n^2}-\frac {(d+e x) (f+g x)^3}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[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*} \text {integral}& = -\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1674\) vs. \(2(339)=678\).

Time = 0.46 (sec) , antiderivative size = 1674, normalized size of antiderivative = 4.94 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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}} \operatorname {ExpIntegralEi}\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}} \operatorname {ExpIntegralEi}\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 \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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} \operatorname {ExpIntegralEi}\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}} \operatorname {ExpIntegralEi}\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}} \operatorname {ExpIntegralEi}\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 \operatorname {ExpIntegralEi}\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 )} \]

[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]))

Maple [C] (warning: unable to verify)

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

Time = 1.70 (sec) , antiderivative size = 9517, normalized size of antiderivative = 28.07

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

[In]

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

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.01 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {{\left (9 \, {\left (a e f g^{2} - a d g^{3} + {\left (b e f g^{2} - b d g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e f g^{2} - b d g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 6 \, {\left (a e^{2} f^{2} g - 2 \, a d e f g^{2} + a d^{2} g^{3} + {\left (b e^{2} f^{2} g - 2 \, b d e f g^{2} + b d^{2} g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e^{2} f^{2} g - 2 \, b d e f g^{2} + b d^{2} g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + {\left (a e^{3} f^{3} - 3 \, a d e^{2} f^{2} g + 3 \, a d^{2} e f g^{2} - a d^{3} g^{3} + {\left (b e^{3} f^{3} - 3 \, b d e^{2} f^{2} g + 3 \, b d^{2} e f g^{2} - b d^{3} g^{3}\right )} n \log \left (e x + d\right ) + {\left (b e^{3} f^{3} - 3 \, b d e^{2} f^{2} g + 3 \, b d^{2} e f g^{2} - b d^{3} g^{3}\right )} \log \left (c\right )\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) - {\left (b e^{4} g^{3} n x^{4} + b d e^{3} f^{3} n + {\left (3 \, b e^{4} f g^{2} + b d e^{3} g^{3}\right )} n x^{3} + 3 \, {\left (b e^{4} f^{2} g + b d e^{3} f g^{2}\right )} n x^{2} + {\left (b e^{4} f^{3} + 3 \, b d e^{3} f^{2} g\right )} n x\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 4 \, {\left (b g^{3} n \log \left (e x + d\right ) + b g^{3} \log \left (c\right ) + a g^{3}\right )} \operatorname {log\_integral}\left ({\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} e^{4} n^{3} \log \left (e x + d\right ) + b^{3} e^{4} n^{2} \log \left (c\right ) + a b^{2} e^{4} n^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3473 vs. \(2 (341) = 682\).

Time = 0.42 (sec) , antiderivative size = 3473, normalized size of antiderivative = 10.24 \[ \int \frac {(f+g x)^3}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)