3.4.18 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{x^2 (f+g x^2)} \, dx\) [318]

Optimal. Leaf size=461 \[ \frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}+\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.44, antiderivative size = 461, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2463, 2444, 2441, 2352, 2456, 2443, 2481, 2421, 6724} \begin {gather*} -\frac {b \sqrt {g} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{3/2}}+\frac {b^2 \sqrt {g} n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d f}+\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}-\frac {\sqrt {g} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (-f)^{3/2}}+\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*e*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f*x)
+ (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2
)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(
3/2)) - (b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(
-f)^(3/2) + (b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/
(-f)^(3/2) + (2*b^2*e*n^2*PolyLog[2, 1 + (e*x)/d])/(d*f) + (b^2*Sqrt[g]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(
e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(3/2) - (b^2*Sqrt[g]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g
])])/(-f)^(3/2)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2441

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

Rule 2443

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

Rule 2444

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[(d + e
*x)*((a + b*Log[c*(d + e*x)^n])^p/((e*f - d*g)*(f + g*x))), x] - Dist[b*e*n*(p/(e*f - d*g)), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 \left (f+g x^2\right )} \, dx &=\int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2} \, dx}{f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}-\frac {g \int \left (\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f}+\frac {(2 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{d f}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 (-f)^{3/2}}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 (-f)^{3/2}}-\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d f}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}-\frac {\left (b e \sqrt {g} n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{(-f)^{3/2}}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{(-f)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}-\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}+d \sqrt {g}}{e}-\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{3/2}}+\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}-d \sqrt {g}}{e}+\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}+\frac {\left (b^2 \sqrt {g} n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{3/2}}-\frac {\left (b^2 \sqrt {g} n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{(-f)^{3/2}}\\ &=\frac {2 b e n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d f}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d f x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2}}-\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b \sqrt {g} n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {2 b^2 e n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d f}+\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}-\frac {b^2 \sqrt {g} n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.27, size = 668, normalized size = 1.45 \begin {gather*} \frac {-2 d \sqrt {f} \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 d \sqrt {g} x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (2 \sqrt {f} (e x \log (x)-(d+e x) \log (d+e x))+i d \sqrt {g} x \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\text {Li}_2\left (-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )-i d \sqrt {g} x \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )\right )+b^2 n^2 \left (2 \sqrt {f} \left (2 e x \log \left (-\frac {e x}{d}\right ) \log (d+e x)-(d+e x) \log ^2(d+e x)+2 e x \text {Li}_2\left (1+\frac {e x}{d}\right )\right )-i d \sqrt {g} x \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+i d \sqrt {g} x \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{2 d f^{3/2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*Sqrt[f]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 2*d*Sqrt[g]*x*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a -
b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(2*Sqrt[f]*(
e*x*Log[x] - (d + e*x)*Log[d + e*x]) + I*d*Sqrt[g]*x*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f]
- I*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) - I*d*Sqrt[g]*x*(Log[d + e*x
]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I
*d*Sqrt[g])])) + b^2*n^2*(2*Sqrt[f]*(2*e*x*Log[-((e*x)/d)]*Log[d + e*x] - (d + e*x)*Log[d + e*x]^2 + 2*e*x*Pol
yLog[2, 1 + (e*x)/d]) - I*d*Sqrt[g]*x*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])
] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d + e
*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) + I*d*Sqrt[g]*x*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] +
d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g
]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])))/(2*d*f^(3/2)*x)

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{x^{2} \left (g \,x^{2}+f \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^2\,\left (g\,x^2+f\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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