3.617 \(\int \frac {a+b \log (c (d+e x^2)^p)}{\sqrt {h x} (f+g x)} \, dx\)

Optimal. Leaf size=1361 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}} \]

[Out]

2*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*(a+b*ln(c*(e*x^2+d)^p))/f^(1/2)/g^(1/2)/h^(1/2)+8*b*p*arctan(g^(
1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*h^(1/2)/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2
)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2
)-e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h
*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(-2*f^(1/2)*g^(1/2)*((
-d)^(1/4)*h^(1/2)-e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)-(-d)^(1/4)*g^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)
^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2
)*((-d)^(1/4)*(-h)^(1/2)+e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)+I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/
2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-2*b*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*l
n(2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)+e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)+(-d)^(1/4)*g^(1/2))/(f^(1/2)*h
^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)-4*I*b*p*polylog(2,1-2*f^(1/2)*h^(1/2)/(f^(1/2)*h^(1/2)-
I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+I*b*p*polylog(2,1-2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(
1/2)-e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)
*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+I*b*p*polylog(2,1+2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)-e^(1/4)*(h*x)^(
1/2))/(I*e^(1/4)*f^(1/2)-(-d)^(1/4)*g^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+
I*b*p*polylog(2,1-2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2)+e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-
h)^(1/2)+I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)+I*b*p*pol
ylog(2,1-2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)+e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)+(-d)^(1/4)*g^(1/2))/(f^
(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/f^(1/2)/g^(1/2)/h^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.83, antiderivative size = 1361, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2467, 205, 2470, 12, 260, 6725, 4928, 4856, 2402, 2315, 2447} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2)^p]))/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (8*
b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*
x])])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*
Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*
(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt
[f]*Sqrt[h])]*Log[(-2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/
4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt
[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/
4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*S
qrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^
(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f
]*Sqrt[g]*Sqrt[h]) - ((4*I)*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(
Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqr
t[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])
/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x])
)/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h
]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*
Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt
[h]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f]
+ (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h])

Rule 12

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

Rule 205

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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]

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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2467

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r
_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + (g*x^k)/h)^r*(a + b*Lo
g[c*(d + (e*x^(k*n))/h^n)^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && Fract
ionQ[m] && IntegerQ[n] && IntegerQ[r]

Rule 2470

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

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{f+\frac {g x^2}{h}} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \operatorname {Subst}\left (\int \frac {\sqrt {h} x^3 \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt {f} \sqrt {g} \left (d+\frac {e x^4}{h^2}\right )} \, dx,x,\sqrt {h x}\right )}{h^3}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \operatorname {Subst}\left (\int \frac {x^3 \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} h^{5/2}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \operatorname {Subst}\left (\int \left (\frac {h^2 x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 \left (-\sqrt {-d} \sqrt {e} h+e x^2\right )}+\frac {h^2 x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 \left (\sqrt {-d} \sqrt {e} h+e x^2\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} h^{5/2}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{-\sqrt {-d} \sqrt {e} h+e x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt {-d} \sqrt {e} h+e x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \operatorname {Subst}\left (\int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \operatorname {Subst}\left (\int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {\left (2 b \sqrt [4]{e} p\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {\left (2 b \sqrt [4]{e} p\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {\left (2 b \sqrt [4]{e} p\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {\left (2 b \sqrt [4]{e} p\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-4 \frac {(2 b p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x\right )}{\sqrt {f} \left (-i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g} \sqrt {-h}}{\sqrt {f} \sqrt {h}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x\right )}{\sqrt {f} \left (-i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g}}{\sqrt {f}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x\right )}{\sqrt {f} \left (i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g} \sqrt {-h}}{\sqrt {f} \sqrt {h}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x\right )}{\sqrt {f} \left (i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g}}{\sqrt {f}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-4 \frac {(2 i b p) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {Li}_2\left (1-\frac {2}{1-\frac {i \sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 1297, normalized size = 0.95 \[ \frac {\sqrt {x} \left (a \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{-d} \sqrt {g}-\sqrt [4]{e} \sqrt {-f}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (i \sqrt [4]{e} \sqrt {x}+\sqrt [4]{-d}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{e} \sqrt {x}+i \sqrt [4]{-d}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{e} \sqrt {x}+\sqrt [4]{-d}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )+b \log \left (c \left (e x^2+d\right )^p\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-a \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (i \sqrt [4]{e} \sqrt {x}+\sqrt [4]{-d}\right )}{\sqrt [4]{-d} \sqrt {g}-i \sqrt [4]{e} \sqrt {-f}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{e} \sqrt {x}+\sqrt [4]{-d}\right )}{\sqrt [4]{-d} \sqrt {g}-\sqrt [4]{e} \sqrt {-f}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )-b \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right ) \log \left (c \left (e x^2+d\right )^p\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )\right )}{\sqrt {-f} \sqrt {g} \sqrt {h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(a*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt
[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x
]))/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sqrt[g]*(I*(-d)^(1/4
) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] - b*p*Log[(Sq
rt[g]*((-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]
- a*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-
d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4) - I*e^(1/4)*Sqrt[x]))/(I*e^(
1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4
)*Sqrt[x]))/((-I)*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b*p*Log[(Sqrt[g]*(
(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]] + b
*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*Log[c*(d + e*x^2)^p] - b*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*Log[c*(d + e*x^2)^p]
- b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] - b*p*PolyLo
g[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] - b*p*PolyLog[2, (e^(1/
4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] - b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f
] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])] + b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*S
qrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] + b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(
1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-
f] + I*(-d)^(1/4)*Sqrt[g])] + b*p*PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(
1/4)*Sqrt[g])]))/(Sqrt[-f]*Sqrt[g]*Sqrt[h*x])

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fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {h x} b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + \sqrt {h x} a}{g h x^{2} + f h x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a}{{\left (g x + f\right )} \sqrt {h x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.91, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )+a}{\sqrt {h x}\, \left (g x +f \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\sqrt {h} p \log \left (e x^{2} + d\right ) + \sqrt {h} \log \relax (c)}{g h x^{\frac {3}{2}} + f h \sqrt {x}}\,{d x} + \frac {2 \, a \arctan \left (\frac {\sqrt {h x} g}{\sqrt {f g h}}\right )}{\sqrt {f g h}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate((sqrt(h)*p*log(e*x^2 + d) + sqrt(h)*log(c))/(g*h*x^(3/2) + f*h*sqrt(x)), x) + 2*a*arctan(sqrt(h*x)
*g/sqrt(f*g*h))/sqrt(f*g*h)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{\left (f+g\,x\right )\,\sqrt {h\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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