\(\int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx\) [206]

Optimal result

Integrand size = 23, antiderivative size = 242 \[ \int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2458, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352} \[ \int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}} \]

[In]

Int[Log[a + b*x]/((a + b*x)*Sqrt[d + e*x]),x]

[Out]

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

Rule 12

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

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 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

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 2390

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.))/(x_), x_Symbol] :> With[{u = IntHi
de[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IntegerQ[q - 1/2]

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 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.52 \[ \int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx=\frac {2 \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-\log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-2 \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+\log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )-2 \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{2 \sqrt {b} \sqrt {b d-a e}} \]

[In]

Integrate[Log[a + b*x]/((a + b*x)*Sqrt[d + e*x]),x]

[Out]

(2*Log[a + b*x]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]] - Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]]^2
- 2*Log[a + b*x]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]] + Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]^2
+ 2*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]]*Log[1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])] - 2*L
og[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]]*Log[(1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])/2] - 2*PolyLog[2
, 1/2 - (Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])] + 2*PolyLog[2, (1 + (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*
e])/2])/(2*Sqrt[b]*Sqrt[b*d - a*e])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.02 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {2 \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a e -b d}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a e -b d}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{2 b}\)	\(155\)
default	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {2 \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a e -b d}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a e -b d}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{2 b}\)	\(155\)

[In]

int(ln(b*x+a)/(b*x+a)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/b*sum(1/_alpha*(2*ln((e*x+d)^(1/2)-_alpha)*ln(((e*x+d)*b+a*e-b*d)/e)-b*(1/_alpha/b*ln((e*x+d)^(1/2)-_alpha
)^2+2*_alpha/(a*e-b*d)*ln((e*x+d)^(1/2)-_alpha)*ln(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)+2*_alpha/(a*e-b*d)*dilog
(1/2*((e*x+d)^(1/2)+_alpha)/_alpha))),_alpha=RootOf(_Z^2*b+a*e-b*d))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(log(a + b*x)/((a + b*x)*sqrt(d + e*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(log(b*x+a)/(b*x+a)/(e*x+d)^(1/2),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(a*e-b*d>0)', see `assume?` for
more detail

Giac [F]

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

[In]

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

[Out]

integrate(log(b*x + a)/((b*x + a)*sqrt(e*x + d)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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