\(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {2+g x^2}} \, dx\) [275]

Optimal result

Integrand size = 26, antiderivative size = 326 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 \sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}+\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{\sqrt {g}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {221, 2451, 12, 5827, 5680, 2221, 2317, 2438} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {g d^2+2 e^2}}\right )}{\sqrt {g}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{\sqrt {g} d+\sqrt {g d^2+2 e^2}}\right )}{\sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {d^2 g+2 e^2}}+1\right )}{\sqrt {g}}-\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{\sqrt {d^2 g+2 e^2}+d \sqrt {g}}+1\right )}{\sqrt {g}}+\frac {b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )^2}{2 \sqrt {g}} \]

[In]

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

[Out]

(b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]^2)/(2*Sqrt[g]) - (b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*e*E^ArcS
inh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + d^2*g])])/Sqrt[g] - (b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Log[
1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g])])/Sqrt[g] + (ArcSinh[(Sqrt[g]
*x)/Sqrt[2]]*(a + b*Log[c*(d + e*x)^n]))/Sqrt[g] - (b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]]
)/(d*Sqrt[g] - Sqrt[2*e^2 + d^2*g]))])/Sqrt[g] - (b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/
(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g]))])/Sqrt[g]

Rule 12

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

Rule 221

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 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 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5827

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2+g x^2}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right ) \left (2 a+b n \text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )-2 b n \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}-\sqrt {2 e^2+d^2 g}}\right )-2 b n \log \left (1+\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )+2 b \log \left (c (d+e x)^n\right )\right )-2 b n \operatorname {PolyLog}\left (2,\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{-d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )-2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e e^{\text {arcsinh}\left (\frac {\sqrt {g} x}{\sqrt {2}}\right )}}{d \sqrt {g}+\sqrt {2 e^2+d^2 g}}\right )}{2 \sqrt {g}} \]

[In]

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

[Out]

(ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*(2*a + b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]] - 2*b*n*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sq
rt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + d^2*g])] - 2*b*n*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]
])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g])] + 2*b*Log[c*(d + e*x)^n]) - 2*b*n*PolyLog[2, (Sqrt[2]*e*E^ArcSinh[(Sqrt[
g]*x)/Sqrt[2]])/(-(d*Sqrt[g]) + Sqrt[2*e^2 + d^2*g])] - 2*b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sq
rt[2]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g]))])/(2*Sqrt[g])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

b*integrate((log((e*x + d)^n) + log(c))/sqrt(g*x^2 + 2), x) + a*arcsinh(1/2*sqrt(2)*sqrt(g)*x)/sqrt(g)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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