3.270 \(\int \frac{(d+e x^2)^{3/2} (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=400 \[ -\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 \sqrt{\frac{e x^2}{d}+1}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{3}{2} e x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 \sqrt{d} \sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{\frac{e x^2}{d}+1}}-\frac{1}{4} b e n x \sqrt{d+e x^2}-\frac{b d n \sqrt{d+e x^2}}{x}+\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 \sqrt{\frac{e x^2}{d}+1}}+\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{\frac{e x^2}{d}+1}}-\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{\frac{e x^2}{d}+1}} \]

[Out]

-((b*d*n*Sqrt[d + e*x^2])/x) - (b*e*n*x*Sqrt[d + e*x^2])/4 + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(S
qrt[e]*x)/Sqrt[d]])/(4*Sqrt[1 + (e*x^2)/d]) + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[
d]]^2)/(4*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E
^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[1 + (e*x^2)/d]) + (3*e*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/2 - (
(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x + (3*Sqrt[d]*Sqrt[e]*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a +
b*Log[c*x^n]))/(2*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt
[e]*x)/Sqrt[d]])])/(4*Sqrt[1 + (e*x^2)/d])

________________________________________________________________________________________

Rubi [A]  time = 0.480609, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2341, 277, 195, 215, 2350, 12, 14, 5659, 3716, 2190, 2279, 2391} \[ -\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 \sqrt{\frac{e x^2}{d}+1}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{3}{2} e x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac{3 \sqrt{d} \sqrt{e} \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{\frac{e x^2}{d}+1}}-\frac{1}{4} b e n x \sqrt{d+e x^2}-\frac{b d n \sqrt{d+e x^2}}{x}+\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 \sqrt{\frac{e x^2}{d}+1}}+\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 \sqrt{\frac{e x^2}{d}+1}}-\frac{3 b \sqrt{d} \sqrt{e} n \sqrt{d+e x^2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x^2,x]

[Out]

-((b*d*n*Sqrt[d + e*x^2])/x) - (b*e*n*x*Sqrt[d + e*x^2])/4 + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(S
qrt[e]*x)/Sqrt[d]])/(4*Sqrt[1 + (e*x^2)/d]) + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[
d]]^2)/(4*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E
^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[1 + (e*x^2)/d]) + (3*e*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/2 - (
(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x + (3*Sqrt[d]*Sqrt[e]*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a +
b*Log[c*x^n]))/(2*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt
[e]*x)/Sqrt[d]])])/(4*Sqrt[1 + (e*x^2)/d])

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[(d^IntPart[
q]*(d + e*x^2)^FracPart[q])/(1 + (e*x^2)/d)^FracPart[q], Int[x^m*(1 + (e*x^2)/d)^q*(a + b*Log[c*x^n]), x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[m/2] && IntegerQ[q - 1/2] &&  !(LtQ[m + 2*q, -2] || GtQ[d, 0])

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5659

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

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]

Rubi steps

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

Mathematica [C]  time = 0.991153, size = 329, normalized size = 0.82 \[ -\frac{b \sqrt{d} n \sqrt{d+e x^2} \left (\sqrt{d} \, _3F_2\left (-\frac{1}{2},-\frac{1}{2},-\frac{1}{2};\frac{1}{2},\frac{1}{2};-\frac{e x^2}{d}\right )+\log (x) \left (\sqrt{d} \sqrt{\frac{e x^2}{d}+1}-\sqrt{e} x \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )\right )}{x \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{e} n \sqrt{d+e x^2} \left ((2 \log (x)-1) \left (\sqrt{e} x \sqrt{\frac{e x^2}{d}+1}+\sqrt{d} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )-2 \sqrt{e} x \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{e x^2}{d}\right )\right )}{4 \sqrt{\frac{e x^2}{d}+1}}-\frac{\left (2 d-e x^2\right ) \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 x}+\frac{3}{2} d \sqrt{e} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x^2,x]

[Out]

-((b*Sqrt[d]*n*Sqrt[d + e*x^2]*(Sqrt[d]*HypergeometricPFQ[{-1/2, -1/2, -1/2}, {1/2, 1/2}, -((e*x^2)/d)] + (Sqr
t[d]*Sqrt[1 + (e*x^2)/d] - Sqrt[e]*x*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])*Log[x]))/(x*Sqrt[1 + (e*x^2)/d])) + (b*Sqrt
[e]*n*Sqrt[d + e*x^2]*(-2*Sqrt[e]*x*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -((e*x^2)/d)] + (Sqrt[e]*x*
Sqrt[1 + (e*x^2)/d] + Sqrt[d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])*(-1 + 2*Log[x])))/(4*Sqrt[1 + (e*x^2)/d]) - ((2*d
- e*x^2)*Sqrt[d + e*x^2]*(a - b*n*Log[x] + b*Log[c*x^n]))/(2*x) + (3*d*Sqrt[e]*(a - b*n*Log[x] + b*Log[c*x^n])
*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/2

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Maple [F]  time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x^2+d)^(3/2)*(a+b*ln(c*x^n))/x^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((e*x^2+d)^(3/2)*(a+b*log(c*x^n))/x^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{{\left (b e x^{2} + b d\right )} \sqrt{e x^{2} + d} \log \left (c x^{n}\right ) +{\left (a e x^{2} + a d\right )} \sqrt{e x^{2} + d}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))*(d + e*x**2)**(3/2)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)^(3/2)*(b*log(c*x^n) + a)/x^2, x)