3.312 \(\int \frac{a+b \log (c x^n)}{x^3 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=489 \[ -\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

-(b*n*(d^2 - e^2*x^2))/(4*d^2*x^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt
[1 - (e^2*x^2)/d^2]])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 -
(e^2*x^2)/d^2]]^2)/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(2*d^2*x^2*Sqrt[
d - e*x]*Sqrt[d + e*x]) - (e^2*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*(a + b*Log[c*x^n]))/(2
*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*Log[2/(1
- Sqrt[1 - (e^2*x^2)/d^2])])/(2*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2
, -((1 + Sqrt[1 - (e^2*x^2)/d^2])/(1 - Sqrt[1 - (e^2*x^2)/d^2]))])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.724349, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2342, 266, 51, 63, 208, 2350, 47, 5984, 5918, 2402, 2315} \[ -\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}+1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*n*(d^2 - e^2*x^2))/(4*d^2*x^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt
[1 - (e^2*x^2)/d^2]])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 -
(e^2*x^2)/d^2]]^2)/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(2*d^2*x^2*Sqrt[
d - e*x]*Sqrt[d + e*x]) - (e^2*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*(a + b*Log[c*x^n]))/(2
*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*Log[2/(1
- Sqrt[1 - (e^2*x^2)/d^2])])/(2*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2
, -((1 + Sqrt[1 - (e^2*x^2)/d^2])/(1 - Sqrt[1 - (e^2*x^2)/d^2]))])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(q_)*((d2_) + (e2_.)*(x_))^(q_), x_
Symbol] :> Dist[((d1 + e1*x)^q*(d2 + e2*x)^q)/(1 + (e1*e2*x^2)/(d1*d2))^q, Int[x^m*(1 + (e1*e2*x^2)/(d1*d2))^q
*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m]
&& IntegerQ[q - 1/2]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 5984

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5918

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

Rule 2402

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

Rule 2315

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

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{a+b \log \left (c x^n\right )}{x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \left (-\frac{\sqrt{1-\frac{e^2 x^2}{d^2}}}{2 x^3}-\frac{e^2 \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 x}\right ) \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\sqrt{1-\frac{e^2 x^2}{d^2}}}{x^3} \, dx}{2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{x} \, dx}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{e^2 x}{d^2}}}{x^2} \, dx,x,x^2\right )}{4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\sqrt{1-\frac{e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{e^2 x}{d^2}}} \, dx,x,x^2\right )}{8 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{d^2 x^2}{e^2}} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \log \left (\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b e^2 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{Li}_2\left (1-\frac{2}{1-\sqrt{1-\frac{e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 0.858847, size = 255, normalized size = 0.52 \[ \frac{\frac{b n \left (e^2 x^2-d^2\right ) \left (2 d^3 \, _3F_2\left (\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};\frac{d^2}{e^2 x^2}\right )+9 e^2 x^2 (2 \log (x)+1) \left (d \sqrt{1-\frac{d^2}{e^2 x^2}}-e x \sin ^{-1}\left (\frac{d}{e x}\right )\right )\right )}{e^2 x^4 \sqrt{1-\frac{d^2}{e^2 x^2}} \sqrt{d-e x} \sqrt{d+e x}}-18 e^2 \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac{18 d \sqrt{d-e x} \sqrt{d+e x} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}+18 e^2 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{36 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*n*(-d^2 + e^2*x^2)*(2*d^3*HypergeometricPFQ[{3/2, 3/2, 3/2}, {5/2, 5/2}, d^2/(e^2*x^2)] + 9*e^2*x^2*(d*Sqr
t[1 - d^2/(e^2*x^2)] - e*x*ArcSin[d/(e*x)])*(1 + 2*Log[x])))/(e^2*Sqrt[1 - d^2/(e^2*x^2)]*x^4*Sqrt[d - e*x]*Sq
rt[d + e*x]) - (18*d*Sqrt[d - e*x]*Sqrt[d + e*x]*(a - b*n*Log[x] + b*Log[c*x^n]))/x^2 + 18*e^2*Log[x]*(a - b*n
*Log[x] + b*Log[c*x^n]) - 18*e^2*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d + Sqrt[d - e*x]*Sqrt[d + e*x]])/(36*d^3
)

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Maple [F]  time = 0.659, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

int((a+b*ln(c*x^n))/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x + d} \sqrt{-e x + d} b \log \left (c x^{n}\right ) + \sqrt{e x + d} \sqrt{-e x + d} a}{e^{2} x^{5} - d^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-(sqrt(e*x + d)*sqrt(-e*x + d)*b*log(c*x^n) + sqrt(e*x + d)*sqrt(-e*x + d)*a)/(e^2*x^5 - d^2*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**3/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{\sqrt{e x + d} \sqrt{-e x + d} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/(sqrt(e*x + d)*sqrt(-e*x + d)*x^3), x)