3.4.27 \(\int \frac {(a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [327]

Optimal. Leaf size=821 \[ -\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.65, antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2456, 2444, 2441, 2440, 2438, 2443, 2481, 2421, 6724} \begin {gather*} -\frac {b^2 e \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n^2}{2 (-f)^{3/2} \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \sqrt {g}}-\frac {b e \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {b \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) n}{2 (-f)^{3/2} \sqrt {g}}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {g} d+e \sqrt {-f}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {g} x+\sqrt {-f}\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 (-f)^{3/2} \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{3/2} \sqrt {g}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(f*(e*Sqrt[-f] + d*Sqrt[g])*(Sqrt[-f] - Sqrt[g]*x)) - ((d + e*x)
*(a + b*Log[c*(d + e*x)^n])^2)/(4*f*(e*Sqrt[-f] - d*Sqrt[g])*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*n*(a + b*Log[c*(d
+ e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) -
((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g])
- (b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(e*(-f)^(3/2
) + d*f*Sqrt[g])*Sqrt[g]) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[
g])])/(4*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(e*(
-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-
f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])
])/(2*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*
Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqr
t[g]))])/(2*(-f)^(3/2)*Sqrt[g]) + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(
3/2)*Sqrt[g])

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 2438

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

Rule 2440

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

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 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}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx\\ &=-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx}{4 f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{-f g-g^2 x^2} \, dx}{2 f}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {g \int \left (-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f g \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 g \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{2 f}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}+g x} \, dx}{2 f \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f} \sqrt {g}-g x} \, dx}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \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 )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b e n \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 )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{3/2}}+\frac {\left (b^2 e^2 n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f} \sqrt {g}-g x\right )}{e \sqrt {-f} \sqrt {g}+d g}\right )}{d+e x} \, dx}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b^2 e^2 n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f} \sqrt {g}+g x\right )}{e \sqrt {-f} \sqrt {g}-d g}\right )}{d+e x} \, dx}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \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 )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \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 )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}+\frac {(b e n) \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}{2 (-f)^{3/2} \sqrt {g}}-\frac {(b e n) \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}{2 (-f)^{3/2} \sqrt {g}}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {g x}{e \sqrt {-f} \sqrt {g}+d g}\right )}{x} \, dx,x,d+e x\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e \sqrt {-f} \sqrt {g}-d g}\right )}{x} \, dx,x,d+e x\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \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 )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \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 )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}+\frac {(b n) \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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {(b n) \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 )}{2 (-f)^{3/2} \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \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 )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \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 )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b 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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b 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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {\left (b^2 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 )}{2 (-f)^{3/2} \sqrt {g}}+\frac {\left (b^2 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 )}{2 (-f)^{3/2} \sqrt {g}}\\ &=-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 f \left (e \sqrt {-f}-d \sqrt {g}\right ) \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e n \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 )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b e n \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 )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {\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 )}{4 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e (-f)^{3/2}+d f \sqrt {g}\right ) \sqrt {g}}+\frac {b 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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 e n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f \left (e \sqrt {-f}+d \sqrt {g}\right ) \sqrt {g}}-\frac {b 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 )}{2 (-f)^{3/2} \sqrt {g}}-\frac {b^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}+\frac {b^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{3/2} \sqrt {g}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.91, size = 1143, normalized size = 1.39 \begin {gather*} \frac {\frac {2 \sqrt {f} x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}+\frac {2 \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}{\sqrt {g}}+\frac {2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {\sqrt {f} \left (\sqrt {g} (d+e x) \log (d+e x)+i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {\sqrt {f} \left (\sqrt {g} (d+e x) \log (d+e x)+e \left (-i \sqrt {f}-\sqrt {g} x\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-i \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 \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 )}{\sqrt {g}}+\frac {b^2 n^2 \left (-\frac {\sqrt {f} \left (-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+\frac {\sqrt {f} \left (\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+i \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 \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 )}{\sqrt {g}}}{4 f^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[f]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a
- b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/Sqrt[g] + (2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*
((Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + I*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sqrt[f] - Sqrt[g]*x]))/((e*Sqrt[
f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (Sqrt[f]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + e*((-I)*Sqrt[f] - Sqrt
[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - I*(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*(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])])))/Sqrt[g] + (b^2*n^2*(-((Sqrt[f]*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2)
+ 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e*(
I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]))/((e*Sqrt[f] + I*d*Sqrt[g]
)*(Sqrt[f] - I*Sqrt[g]*x))) + (Sqrt[f]*(Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sq
rt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog
[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + I*(
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
*(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])])))/Sqrt[g])
/(4*f^(3/2))

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{\left (g \,x^{2}+f \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*a^2*(x/(f*g*x^2 + f^2) + arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f)) + integrate((b^2*log((x*e + d)^n)^2 + b^2*lo
g(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log((x*e + d)^n))/(g^2*x^4 + 2*f*g*x^2 + f^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/(g*x^2+f)^2,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^2*x^4 + 2*f*g*x^2 + f^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/(g*x**2+f)**2,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/(g*x^2+f)^2,x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)^2/(g*x^2 + f)^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}{{\left (g\,x^2+f\right )}^2} \,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/(f + g*x^2)^2,x)

[Out]

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

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