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

Optimal. Leaf size=242 \[ -\frac{2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{\sqrt{b} \sqrt{b d-a e}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{\sqrt{b} \sqrt{b d-a e}}-\frac{2 \log (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}}-\frac{4 \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]

[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])

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Rubi [A]  time = 0.645323, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {2411, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315} \[ -\frac{2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{\sqrt{b} \sqrt{b d-a e}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{\sqrt{b} \sqrt{b d-a e}}-\frac{2 \log (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}}-\frac{4 \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

[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 2411

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 63

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 208

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

Rule 2348

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 12

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

Rule 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 6741

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

Rule 5984

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 5918

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 2402

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\log (a+b x)}{(a+b x) \sqrt{d+e x}} \, dx &=\frac{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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]  time = 2.69314, size = 239, normalized size = 0.99 \[ \frac{\frac{\sqrt{\frac{b (d+e x)}{b d-a e}} \left (-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{\frac{b (d+e x)}{b d-a e}}\right )+\log ^2\left (\frac{e (a+b x)}{a e-b d}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{\frac{b (d+e x)}{b d-a e}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{\frac{b (d+e x)}{b d-a e}}+1\right )\right ) \log \left (\frac{e (a+b x)}{a e-b d}\right )\right )}{2 \sqrt{d+e x}}-\frac{2 \left (\log (a+b x)-\log \left (\frac{e (a+b x)}{a e-b d}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-\frac{a e}{b}}}\right )}{\sqrt{d-\frac{a e}{b}}}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.842, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( bx+a \right ) }{bx+a}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d} \log \left (b x + a\right )}{b e x^{2} + a d +{\left (b d + a e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right )}{{\left (b x + a\right )} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)