Integrand size = 23, antiderivative size = 211 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 b \sqrt {d} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \]
4*b*n*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)+2*b*n*arctanh((e*x+d)^(1/2)/d ^(1/2))^2*d^(1/2)-2*arctanh((e*x+d)^(1/2)/d^(1/2))*(a+b*ln(c*x^n))*d^(1/2) -4*b*n*arctanh((e*x+d)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(e*x+d)^(1/2)) )*d^(1/2)-2*b*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(e*x+d)^(1/2)))*d^(1/2)-4*b *n*(e*x+d)^(1/2)+2*(a+b*ln(c*x^n))*(e*x+d)^(1/2)
Time = 0.16 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=2 a \sqrt {d+e x}-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d+e x} \log \left (c x^n\right )+\sqrt {d} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-\sqrt {d} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-\frac {1}{2} b \sqrt {d} n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+\frac {1}{2} b \sqrt {d} n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right ) \]
2*a*Sqrt[d + e*x] - 4*b*n*Sqrt[d + e*x] + 4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e *x]/Sqrt[d]] + 2*b*Sqrt[d + e*x]*Log[c*x^n] + Sqrt[d]*(a + b*Log[c*x^n])*L og[Sqrt[d] - Sqrt[d + e*x]] - Sqrt[d]*(a + b*Log[c*x^n])*Log[Sqrt[d] + Sqr t[d + e*x]] - (b*Sqrt[d]*n*(Log[Sqrt[d] - Sqrt[d + e*x]]*(Log[Sqrt[d] - Sq rt[d + e*x]] + 2*Log[(1 + Sqrt[d + e*x]/Sqrt[d])/2]) + 2*PolyLog[2, 1/2 - Sqrt[d + e*x]/(2*Sqrt[d])]))/2 + (b*Sqrt[d]*n*(Log[Sqrt[d] + Sqrt[d + e*x] ]*(Log[Sqrt[d] + Sqrt[d + e*x]] + 2*Log[1/2 - Sqrt[d + e*x]/(2*Sqrt[d])]) + 2*PolyLog[2, (1 + Sqrt[d + e*x]/Sqrt[d])/2]))/2
Time = 1.02 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2788, 2756, 60, 73, 221, 2790, 27, 7267, 25, 6546, 27, 6470, 27, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle e \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}}dx+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \int \frac {\sqrt {d+e x}}{x}dx}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (d \int \frac {1}{x \sqrt {d+e x}}dx+2 \sqrt {d+e x}\right )}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (\frac {2 d \int \frac {1}{\frac {d+e x}{e}-\frac {d}{e}}d\sqrt {d+e x}}{e}+2 \sqrt {d+e x}\right )}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle d \left (-b n \int -\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x}dx-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {2 b n \int \frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x}dx}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle d \left (\frac {4 b n \int \frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (-\frac {4 b n \int -\frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\frac {\int \frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\int \frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle d \left (\frac {4 b n \left (\frac {\int -\frac {d \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {4 b n \left (\sqrt {d} \int -\frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle d \left (\frac {4 b n \left (-\sqrt {d} \int \frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}}d\frac {1}{\sqrt {d}-\sqrt {d+e x}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\) |
e*((-2*b*n*(2*Sqrt[d + e*x] - 2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]))/e + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e) + d*((-2*ArcTanh[Sqrt[d + e*x]/ Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[d] + (4*b*n*(ArcTanh[Sqrt[d + e*x]/Sqrt[ d]]^2/2 - ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] - PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]/2))/Sqrt [d])
3.2.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) /(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L og[c*x^n]), x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, n , r}, x] && IntegerQ[q - 1/2]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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] + Simp[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]
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] + Simp[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]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{x}d x\]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x}}{x}\, dx \]
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
(sqrt(d)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d))) + 2*sqrt (e*x + d))*a + b*integrate(sqrt(e*x + d)*(log(c) + log(x^n))/x, x)
\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x}}{x} \,d x \]