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

Optimal. Leaf size=1361 \[ \text{result too large to display} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.83058, antiderivative size = 1361, normalized size of antiderivative = 1., 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 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 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 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 12

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

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 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]

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 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 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]

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]]

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*}

Mathematica [A]  time = 0.387796, 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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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{PolyLog}\left (2,\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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Maple [F]  time = 1.306, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) }{gx+f}{\frac{1}{\sqrt{hx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*(e*x^2+d)^p))/(h*x)^(1/2)/(g*x+f),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((a+b*log(c*(e*x^2+d)^p))/(h*x)^(1/2)/(g*x+f),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{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 ) \end{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)