\(\int \frac {\sqrt {d+e x^2} (a+b \log (c x^n))}{x^3} \, dx\) [255]

Optimal result

Integrand size = 25, antiderivative size = 252 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b n \sqrt {d+e x^2}}{4 x^2}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{4 \sqrt {d}}+\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{4 \sqrt {d}}-\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 \sqrt {d}}-\frac {b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{4 \sqrt {d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {272, 43, 65, 214, 2392, 12, 14, 6131, 6055, 2449, 2352} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {d}}-\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2}{4 \sqrt {d}}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{4 \sqrt {d}}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )}{2 \sqrt {d}}-\frac {b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )}{4 \sqrt {d}}-\frac {b n \sqrt {d+e x^2}}{4 x^2} \]

[In]

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

[Out]

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

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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

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 214

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

Rule 272

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 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 2392

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 2449

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 6055

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 6131

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]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.30 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {-2 b \sqrt {d} n \sqrt {d+e x^2} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {d}{e x^2}\right )-b \sqrt {e} n x \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right ) (1+2 \log (x))+\sqrt {1+\frac {d}{e x^2}} \left (-2 a \sqrt {d} \sqrt {d+e x^2}-b \sqrt {d} n \sqrt {d+e x^2}-2 b e n x^2 \log ^2(x)-2 a e x^2 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )+2 e x^2 \log (x) \left (a+b \log \left (c x^n\right )+b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )-2 b \log \left (c x^n\right ) \left (\sqrt {d} \sqrt {d+e x^2}+e x^2 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )\right )}{4 \sqrt {d} \sqrt {1+\frac {d}{e x^2}} x^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]

[In]

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

[Out]

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