3.618 \(\int \frac{a+b \log (c (d+e x^2)^p)}{(h x)^{3/2} (f+g x)} \, dx\)

Optimal. Leaf size=1659 \[ \text{result too large to display} \]

[Out]

(-2*Sqrt[2]*b*e^(1/4)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) + (2*Sq
rt[2]*b*e^(1/4)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) - (2*(a + b*L
og[c*(d + e*x^2)^p]))/(f*h*Sqrt[h*x]) - (2*Sqrt[g]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*
(d + e*x^2)^p]))/(f^(3/2)*h^(3/2)) + (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^
(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x +
Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (8*b*Sqrt[g]*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])])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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)*S
qrt[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]))
])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + ((4*I)*b*Sqrt[g]*p*PolyLog[
2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLo
g[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]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*Poly
Log[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]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sq
rt[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]*Sqr
t[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 2.52854, antiderivative size = 1659, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 19, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.613, Rules used = {2467, 2476, 2455, 297, 1162, 617, 204, 1165, 628, 205, 2470, 12, 260, 6725, 4928, 4856, 2402, 2315, 2447} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^2)^p])/((h*x)^(3/2)*(f + g*x)),x]

[Out]

(-2*Sqrt[2]*b*e^(1/4)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) + (2*Sq
rt[2]*b*e^(1/4)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) - (2*(a + b*L
og[c*(d + e*x^2)^p]))/(f*h*Sqrt[h*x]) - (2*Sqrt[g]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*
(d + e*x^2)^p]))/(f^(3/2)*h^(3/2)) + (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^
(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x +
Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (8*b*Sqrt[g]*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])])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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)*S
qrt[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]))
])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2)) + ((4*I)*b*Sqrt[g]*p*PolyLog[
2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLo
g[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]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*Poly
Log[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]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sq
rt[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]*Sqr
t[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*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]))])/(f^(3/2)*h^(3/2))

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 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

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

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 )}{(h x)^{3/2} (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 )}{x^2 \left (f+\frac{g x^2}{h}\right )} \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{f x^2}-\frac{g \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{f \left (f h+g x^2\right )}\right ) \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^2} \, dx,x,\sqrt{h x}\right )}{f h}-\frac{(2 g) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{f h+g x^2} \, dx,x,\sqrt{h x}\right )}{f h}\\ &=-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{(8 b e p) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{f h^3}+\frac{(8 b e g p) \operatorname{Subst}\left (\int \frac{x^3 \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f} \sqrt{h}}\right )}{\sqrt{f} \sqrt{g} \sqrt{h} \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{f h^3}\\ &=-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\left (8 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{7/2}}-\frac{\left (4 b \sqrt{e} p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h-\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{f h^3}+\frac{\left (4 b \sqrt{e} p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h+\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{f h^3}\\ &=-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\left (8 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{7/2}}+\frac{\left (\sqrt{2} b \sqrt [4]{e} p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}+2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{\left (\sqrt{2} b \sqrt [4]{e} p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}-2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{(2 b p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{f h}+\frac{(2 b p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{f h}\\ &=-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{\left (2 \sqrt{2} b \sqrt [4]{e} p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\left (2 \sqrt{2} b \sqrt [4]{e} p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{\left (4 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{3/2}}+\frac{\left (4 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{3/2}}\\ &=-\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{\left (4 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{3/2}}+\frac{\left (4 b e \sqrt{g} p\right ) \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 )}{f^{3/2} h^{3/2}}\\ &=-\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\left (2 b \sqrt [4]{e} \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{\left (2 b \sqrt [4]{e} \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{\left (2 b \sqrt [4]{e} \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{\left (2 b \sqrt [4]{e} \sqrt{g} 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 )}{f^{3/2} h^{3/2}}\\ &=-\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{8 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+4 \frac{(2 b g 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^2 h^2}-\frac{(2 b g 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^2 h^2}-\frac{(2 b g 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^2 h^2}-\frac{(2 b g 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^2 h^2}-\frac{(2 b g 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^2 h^2}\\ &=-\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{8 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+4 \frac{\left (2 i b \sqrt{g} p\right ) \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 )}{f^{3/2} h^{3/2}}\\ &=-\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f h \sqrt{h x}}-\frac{2 \sqrt{g} \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 )}{f^{3/2} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{\sqrt [4]{d} f h^{3/2}}-\frac{8 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{2 b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}-\frac{i b \sqrt{g} 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 )}{f^{3/2} h^{3/2}}+\frac{4 i b \sqrt{g} p \text{Li}_2\left (1-\frac{2}{1-\frac{i \sqrt{g} \sqrt{h x}}{\sqrt{f} \sqrt{h}}}\right )}{f^{3/2} h^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.63769, size = 1336, normalized size = 0.81 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^2)^p])/((h*x)^(3/2)*(f + g*x)),x]

[Out]

(x^(3/2)*((4*b*e^(1/4)*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d
)^(1/4) - (2*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x] + (f*Sqrt[g]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(
d + e*x^2)^p]))/(-f)^(3/2) + (Sqrt[g]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[-f] +
(b*Sqrt[g]*p*(Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sq
rt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sq
rt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + 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]] + 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]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt
[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sq
rt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)
^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])])
)/Sqrt[-f] + (b*f*Sqrt[g]*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]] + 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]] + 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]] + 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]] + PolyLog[2, (e^(1/4)*(Sqrt[-
f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt
[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*
Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] + Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)
^(1/4)*Sqrt[g])]))/(-f)^(3/2)))/(f*(h*x)^(3/2))

________________________________________________________________________________________

Maple [F]  time = 1.321, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) }{gx+f} \left ( hx \right ) ^{-{\frac{3}{2}}}}\, 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)^(3/2)/(g*x+f),x)

[Out]

int((a+b*ln(c*(e*x^2+d)^p))/(h*x)^(3/2)/(g*x+f),x)

________________________________________________________________________________________

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)^(3/2)/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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^{2} x^{3} + f h^{2} x^{2}}, 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)^(3/2)/(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^2*x^3 + f*h^2*x^2), x)

________________________________________________________________________________________

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)**(3/2)/(g*x+f),x)

[Out]

Timed out

________________________________________________________________________________________

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 )} \left (h x\right )^{\frac{3}{2}}}\,{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)^(3/2)/(g*x+f),x, algorithm="giac")

[Out]

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