\(\int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx\) [207]

Optimal result

Integrand size = 23, antiderivative size = 316 \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}} \]

[Out]

4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+b*d)^(3/2)+2*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a
*e+b*d)^(1/2))^2*b^(1/2)/(-a*e+b*d)^(3/2)-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)^(3/2)-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)^(3/2)-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)^
(3/2)+2*ln(b*x+a)/(-a*e+b*d)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356} \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{\sqrt {d+e x} (b d-a e)} \]

[In]

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

[Out]

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

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 2356

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] - Dist[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] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.73 \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=2 \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {\log (a+b x)}{(b d-a e) \sqrt {d+e x}}+\frac {\sqrt {b} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )}{2 (b d-a e)^{3/2}}-\frac {\sqrt {b} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )}{2 (b d-a e)^{3/2}}-\frac {\sqrt {b} \left (\log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+2 \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )}{4 (b d-a e)^{3/2}}+\frac {\sqrt {b} \left (2 \log \left (\frac {\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right ) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+\log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )\right )}{4 (b d-a e)^{3/2}}\right ) \]

[In]

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

[Out]

2*((2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2) + Log[a + b*x]/((b*d - a*e)*
Sqrt[d + e*x]) + (Sqrt[b]*Log[a + b*x]*Log[Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x]])/(2*(b*d - a*e)^(3/2)) - (
Sqrt[b]*Log[a + b*x]*Log[Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]])/(2*(b*d - a*e)^(3/2)) - (Sqrt[b]*(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[(Sqrt[b*d - a*e] +
Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])] + 2*PolyLog[2, (Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b
*d - a*e])]))/(4*(b*d - a*e)^(3/2)) + (Sqrt[b]*(2*Log[(Sqrt[b*d - a*e] - Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d -
a*e])]*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*PolyL
og[2, (Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x])/(2*Sqrt[b*d - a*e])]))/(4*(b*d - a*e)^(3/2)))

Maple [C] (verified)

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

Time = 0.90 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {-\frac {2 \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{\sqrt {e x +d}}+\frac {4 b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{a e -b d}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\left (-\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )}{2 \left (a e -b d \right ) \underline {\hspace {1.25 ex}}\alpha }\right )\right )\)	\(243\)
default	\(\frac {-\frac {2 \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{\sqrt {e x +d}}+\frac {4 b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}}{a e -b d}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\left (-\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )}{2 \left (a e -b d \right ) \underline {\hspace {1.25 ex}}\alpha }\right )\right )\)	\(243\)

[In]

int(ln(b*x+a)/(b*x+a)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2*(-1/(e*x+d)^(1/2)*ln(((e*x+d)*b+a*e-b*d)/e)+2*b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/
2)))/(a*e-b*d)+2*Sum(-1/2*(ln((e*x+d)^(1/2)-_alpha)*ln(((e*x+d)*b+a*e-b*d)/e)-2*b*(1/4/_alpha/b*ln((e*x+d)^(1/
2)-_alpha)^2+1/2*_alpha/(a*e-b*d)*ln((e*x+d)^(1/2)-_alpha)*ln(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)+1/2*_alpha/(a
*e-b*d)*dilog(1/2*((e*x+d)^(1/2)+_alpha)/_alpha)))/(a*e-b*d)/_alpha,_alpha=RootOf(_Z^2*b+a*e-b*d))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (a+b x)}{(a+b x) (d+e x)^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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