\(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^2} \, dx\) [95]

Optimal result

Integrand size = 24, antiderivative size = 259 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {3 e^{-\frac {3 a}{b n}} g^2 (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2447, 2446, 2436, 2337, 2209, 2437, 2347} \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {4 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^3 n^2}+\frac {3 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^3 n^2}-\frac {(d+e x) (f+g x)^2}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[In]

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

[Out]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1015\) vs. \(2(259)=518\).

Time = 0.25 (sec) , antiderivative size = 1015, normalized size of antiderivative = 3.92 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {3 a}{b n}} \left (c (d+e x)^n\right )^{-3/n} \left (-b d e^2 e^{\frac {3 a}{b n}} f^2 n \left (c (d+e x)^n\right )^{3/n}-b e^3 e^{\frac {3 a}{b n}} f^2 n x \left (c (d+e x)^n\right )^{3/n}-2 b d e^2 e^{\frac {3 a}{b n}} f g n x \left (c (d+e x)^n\right )^{3/n}-2 b e^3 e^{\frac {3 a}{b n}} f g n x^2 \left (c (d+e x)^n\right )^{3/n}-b d e^2 e^{\frac {3 a}{b n}} g^2 n x^2 \left (c (d+e x)^n\right )^{3/n}-b e^3 e^{\frac {3 a}{b n}} g^2 n x^3 \left (c (d+e x)^n\right )^{3/n}+a e^2 e^{\frac {2 a}{b n}} f^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )-2 a d e e^{\frac {2 a}{b n}} f g (d+e x) \left (c (d+e x)^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+a d^2 e^{\frac {2 a}{b n}} g^2 (d+e x) \left (c (d+e x)^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+4 a e e^{\frac {a}{b n}} f g (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-4 a d e^{\frac {a}{b n}} g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+3 a g^2 (d+e x)^3 \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )+b e^2 e^{\frac {2 a}{b n}} f^2 (d+e x) \left (c (d+e x)^n\right )^{2/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 )-2 b d e e^{\frac {2 a}{b n}} f g (d+e x) \left (c (d+e x)^n\right )^{2/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^2 e^{\frac {2 a}{b n}} g^2 (d+e x) \left (c (d+e x)^n\right )^{2/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 )+4 b e e^{\frac {a}{b n}} f g (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{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 )-4 b d e^{\frac {a}{b n}} g^2 (d+e x)^2 \left (c (d+e x)^n\right )^{\frac {1}{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 )+3 b g^2 (d+e x)^3 \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 )\right )}{b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \]

[In]

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

[Out]

(-(b*d*e^2*E^((3*a)/(b*n))*f^2*n*(c*(d + e*x)^n)^(3/n)) - b*e^3*E^((3*a)/(b*n))*f^2*n*x*(c*(d + e*x)^n)^(3/n)
- 2*b*d*e^2*E^((3*a)/(b*n))*f*g*n*x*(c*(d + e*x)^n)^(3/n) - 2*b*e^3*E^((3*a)/(b*n))*f*g*n*x^2*(c*(d + e*x)^n)^
(3/n) - b*d*e^2*E^((3*a)/(b*n))*g^2*n*x^2*(c*(d + e*x)^n)^(3/n) - b*e^3*E^((3*a)/(b*n))*g^2*n*x^3*(c*(d + e*x)
^n)^(3/n) + a*e^2*E^((2*a)/(b*n))*f^2*(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])
/(b*n)] - 2*a*d*e*E^((2*a)/(b*n))*f*g*(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])
/(b*n)] + a*d^2*E^((2*a)/(b*n))*g^2*(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(
b*n)] + 4*a*e*E^(a/(b*n))*f*g*(d + e*x)^2*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/
(b*n)] - 4*a*d*E^(a/(b*n))*g^2*(d + e*x)^2*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))
/(b*n)] + 3*a*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)] + b*e^2*E^((2*a)/(b*n))*f^2*
(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d + e*x)^n] - 2*b*d*e*E
^((2*a)/(b*n))*f*g*(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)]*Log[c*(d +
e*x)^n] + b*d^2*E^((2*a)/(b*n))*g^2*(d + e*x)*(c*(d + e*x)^n)^(2/n)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(
b*n)]*Log[c*(d + e*x)^n] + 4*b*e*E^(a/(b*n))*f*g*(d + e*x)^2*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(2*(a + b*Lo
g[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] - 4*b*d*E^(a/(b*n))*g^2*(d + e*x)^2*(c*(d + e*x)^n)^n^(-1)*ExpInt
egralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n] + 3*b*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b
*Log[c*(d + e*x)^n]))/(b*n)]*Log[c*(d + e*x)^n])/(b^2*e^3*E^((3*a)/(b*n))*n^2*(c*(d + e*x)^n)^(3/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 = 1.27 (sec) , antiderivative size = 5089, normalized size of antiderivative = 19.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(5089\)

[In]

int((g*x+f)^2/(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 = 0.31 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.67 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {{\left (4 \, {\left (a e f g - a d g^{2} + {\left (b e f g - b d g^{2}\right )} n \log \left (e x + d\right ) + {\left (b e f g - b d g^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + {\left (a e^{2} f^{2} - 2 \, a d e f g + a d^{2} g^{2} + {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} n \log \left (e x + d\right ) + {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} \log \left (c\right )\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) - {\left (b e^{3} g^{2} n x^{3} + b d e^{2} f^{2} n + {\left (2 \, b e^{3} f g + b d e^{2} g^{2}\right )} n x^{2} + {\left (b e^{3} f^{2} + 2 \, b d e^{2} f g\right )} n x\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 3 \, {\left (b g^{2} n \log \left (e x + d\right ) + b g^{2} \log \left (c\right ) + a g^{2}\right )} \operatorname {log\_integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} e^{3} n^{3} \log \left (e x + d\right ) + b^{3} e^{3} n^{2} \log \left (c\right ) + a b^{2} e^{3} n^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-(e*g^2*x^3 + d*f^2 + (2*e*f*g + d*g^2)*x^2 + (e*f^2 + 2*d*f*g)*x)/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c)
+ a*b*e*n) + integrate((3*e*g^2*x^2 + e*f^2 + 2*d*f*g + 2*(2*e*f*g + d*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. 2031 vs. \(2 (260) = 520\).

Time = 0.39 (sec) , antiderivative size = 2031, normalized size of antiderivative = 7.84 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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