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

Optimal. Leaf size=949 \[ -\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 ) f^2}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 b g \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right ) f}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{\sqrt [4]{e} h^{3/2}}-\frac {16 b g p \sqrt {h x} f}{h^2}+\frac {4 a g \sqrt {h x} f}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} h^{3/2}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}} \]

[Out]

-8/9*b*g^2*p*(h*x)^(3/2)/h^3+2/3*g^2*(h*x)^(3/2)*(a+b*ln(c*(e*x^2+d)^p))/h^3-2*b*e^(1/4)*f^2*p*arctan(1-e^(1/4
)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)-4*b*d^(1/4)*f*g*p*arctan(1-e^(1/4)*2^(1/2)*(h*x
)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)/h^(3/2)-2/3*b*d^(3/4)*g^2*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1
/4)/h^(1/2))*2^(1/2)/e^(3/4)/h^(3/2)+2*b*e^(1/4)*f^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2
^(1/2)/d^(1/4)/h^(3/2)+4*b*d^(1/4)*f*g*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(1/4)
/h^(3/2)+2/3*b*d^(3/4)*g^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(3/4)/h^(3/2)+b*e
^(1/4)*f^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(1/4)/h^(3/2)
-2*b*d^(1/4)*f*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/e^(1/4)/h
^(3/2)+1/3*b*d^(3/4)*g^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/e
^(3/4)/h^(3/2)-b*e^(1/4)*f^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/
2)/d^(1/4)/h^(3/2)+2*b*d^(1/4)*f*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))
*2^(1/2)/e^(1/4)/h^(3/2)-1/3*b*d^(3/4)*g^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*e^(1/4)*2^(1/2)*(h*x
)^(1/2))*2^(1/2)/e^(3/4)/h^(3/2)-2*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(1/2)+4*a*f*g*(h*x)^(1/2)/h^2-16*b*f*g*
p*(h*x)^(1/2)/h^2+4*b*f*g*ln(c*(e*x^2+d)^p)*(h*x)^(1/2)/h^2

________________________________________________________________________________________

Rubi [A]  time = 1.26, antiderivative size = 949, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2467, 2476, 2448, 321, 211, 1165, 628, 1162, 617, 204, 2455, 297} \[ -\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 ) f^2}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 b g \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right ) f}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{\sqrt [4]{e} h^{3/2}}-\frac {16 b g p \sqrt {h x} f}{h^2}+\frac {4 a g \sqrt {h x} f}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} h^{3/2}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*f*g*Sqrt[h*x])/h^2 - (16*b*f*g*p*Sqrt[h*x])/h^2 - (8*b*g^2*p*(h*x)^(3/2))/(9*h^3) - (2*Sqrt[2]*b*e^(1/4)*
f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) - (4*Sqrt[2]*b*d^(1/4)*f*g*
p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) - (2*Sqrt[2]*b*d^(3/4)*g^2*p*Ar
cTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*h^(3/2)) + (2*Sqrt[2]*b*e^(1/4)*f^2*p*ArcT
an[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) + (4*Sqrt[2]*b*d^(1/4)*f*g*p*ArcTan[1
+ (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*d^(3/4)*g^2*p*ArcTan[1 + (
Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*h^(3/2)) + (4*b*f*g*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/
h^2 - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*Sqrt[h*x]) + (2*g^2*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/(3
*h^3) + (Sqrt[2]*b*e^(1/4)*f^2*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)*h^(3/2)) - (2*Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1
/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) + (Sqrt[2]*b*d^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]
*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*f^2*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)*h^(3/2)) + (2*Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h]
+ Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) - (Sqrt[2]*b*d^(3/4)*g^2*p*Log[Sq
rt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*h^(3/2))

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 211

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

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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 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 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 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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 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]

Rubi steps

\begin {align*} \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (f+\frac {g x^2}{h}\right )^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^2} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {2 f g \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h}+\frac {f^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^2}+\frac {g^2 x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h^2}\right ) \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {\left (2 g^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(4 f g) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (2 f^2\right ) \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 )}{h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {(4 b f g) \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right ) \, dx,x,\sqrt {h x}\right )}{h^2}-\frac {\left (8 b e g^2 p\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^5}+\frac {\left (8 b e f^2 p\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}-\frac {(16 b e f g p) \operatorname {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^4}-\frac {\left (4 b \sqrt {e} f^2 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 )}{h^3}+\frac {\left (4 b \sqrt {e} f^2 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 )}{h^3}+\frac {\left (8 b d g^2 p\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^3}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}-\frac {\left (4 b d g^2 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 )}{3 \sqrt {e} h^3}+\frac {\left (4 b d g^2 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 )}{3 \sqrt {e} h^3}+\frac {(16 b d f g p) \operatorname {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {\left (2 b f^2 p\right ) \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 )}{h}+\frac {\left (2 b f^2 p\right ) \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 )}{h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {\left (8 b \sqrt {d} f g 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 )}{h^3}+\frac {\left (8 b \sqrt {d} f g 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 )}{h^3}+\frac {\left (2 \sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {\left (\sqrt {2} b d^{3/4} g^2 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 )}{3 e^{3/4} h^{3/2}}+\frac {\left (\sqrt {2} b d^{3/4} g^2 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 )}{3 e^{3/4} h^{3/2}}+\frac {\left (2 b d g^2 p\right ) \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 )}{3 e h}+\frac {\left (2 b d g^2 p\right ) \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 )}{3 e h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}-\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}+\frac {\left (2 \sqrt {2} b d^{3/4} g^2 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 )}{3 e^{3/4} h^{3/2}}-\frac {\left (2 \sqrt {2} b d^{3/4} g^2 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 )}{3 e^{3/4} h^{3/2}}+\frac {\left (4 b \sqrt {d} f g p\right ) \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 )}{\sqrt {e} h}+\frac {\left (4 b \sqrt {d} f g p\right ) \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 )}{\sqrt {e} h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}+\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}-\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}+\frac {\left (4 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}-\frac {\left (4 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b d^{3/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}+\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} f^2 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} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f g 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]{e} h^{3/2}}-\frac {\sqrt {2} b d^{3/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} h^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.87, size = 436, normalized size = 0.46 \[ \frac {2 x^{3/2} \left (-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}+\frac {1}{3} g^2 x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+2 a f g \sqrt {x}+2 b f g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 b g^2 p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}+\frac {2 b \sqrt [4]{e} f^2 p \left (\tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b f g p \left (\sqrt {2} \sqrt [4]{d} \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )-\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )+2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}+1\right )+8 \sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e}}\right )}{(h x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(2*a*f*g*Sqrt[x] - (2*b*g^2*p*(2*(-d)^(1/4)*e^(3/4)*x^(3/2) - 3*d*ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/
4)] + 3*d*ArcTanh[(e^(1/4)*Sqrt[x])/(-d)^(1/4)]))/(9*(-d)^(1/4)*e^(3/4)) + (2*b*e^(1/4)*f^2*p*(ArcTan[(e^(1/4)
*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]))/(-d)^(1/4) - (b*f*g*p*(8*e^(1/4)*Sqrt[x] + 2
*Sqrt[2]*d^(1/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 2*Sqrt[2]*d^(1/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)
*Sqrt[x])/d^(1/4)] + Sqrt[2]*d^(1/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] - Sqrt[2]*d^(1
/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/e^(1/4) + 2*b*f*g*Sqrt[x]*Log[c*(d + e*x^2)^p
] - (f^2*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x] + (g^2*x^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/3))/(h*x)^(3/2)

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fricas [B]  time = 1.79, size = 2118, normalized size = 2.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(3*h^2*x*sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^
3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log(32*(81*b^3*e^
4*f^8 + 108*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 12*b^3*d^3*e*f^2*g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 +
32*((3*d*e^3*f^2 + d^2*e^2*g^2)*h^5*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 -
60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) - 6*(9*b^2*d*e^3*f^5*g - 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^
3*e*f*g^5)*h^2*p^2)*sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*
b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))) - 3*h^2*x*
sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 +
b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log(32*(81*b^3*e^4*f^8 + 108*b^
3*d*e^3*f^6*g^2 - 1242*b^3*d^2*e^2*f^4*g^4 + 12*b^3*d^3*e*f^2*g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 - 32*((3*d*e^3*
f^2 + d^2*e^2*g^2)*h^5*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*
f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) - 6*(9*b^2*d*e^3*f^5*g - 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^3*e*f*g^5)*h^2
*p^2)*sqrt(-(e*h^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*
g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) + 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))) - 3*h^2*x*sqrt((e*h^3*sq
rt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^
4/(d*e^3*h^6)) - 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log(32*(81*b^3*e^4*f^8 + 108*b^3*d*e^3*f^6*g^2
- 1242*b^3*d^2*e^2*f^4*g^4 + 12*b^3*d^3*e*f^2*g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 + 32*((3*d*e^3*f^2 + d^2*e^2*g
^2)*h^5*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d
^4*g^8)*p^4/(d*e^3*h^6)) + 6*(9*b^2*d*e^3*f^5*g - 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^3*e*f*g^5)*h^2*p^2)*sqrt((e*h
^3*sqrt(-(81*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^
8)*p^4/(d*e^3*h^6)) - 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))) + 3*h^2*x*sqrt((e*h^3*sqrt(-(81*b^4*e^4*
f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^6)) -
12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))*log(32*(81*b^3*e^4*f^8 + 108*b^3*d*e^3*f^6*g^2 - 1242*b^3*d^2*
e^2*f^4*g^4 + 12*b^3*d^3*e*f^2*g^6 + b^3*d^4*g^8)*sqrt(h*x)*p^3 - 32*((3*d*e^3*f^2 + d^2*e^2*g^2)*h^5*sqrt(-(8
1*b^4*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e
^3*h^6)) + 6*(9*b^2*d*e^3*f^5*g - 30*b^2*d^2*e^2*f^3*g^3 + b^2*d^3*e*f*g^5)*h^2*p^2)*sqrt((e*h^3*sqrt(-(81*b^4
*e^4*f^8 - 540*b^4*d*e^3*f^6*g^2 + 918*b^4*d^2*e^2*f^4*g^4 - 60*b^4*d^3*e*f^2*g^6 + b^4*d^4*g^8)*p^4/(d*e^3*h^
6)) - 12*(3*b^2*e*f^3*g + b^2*d*f*g^3)*p^2)/(e*h^3))) + (9*a*f^2 + (4*b*g^2*p - 3*a*g^2)*x^2 + 18*(4*b*f*g*p -
a*f*g)*x - 3*(b*g^2*p*x^2 + 6*b*f*g*p*x - 3*b*f^2*p)*log(e*x^2 + d) - 3*(b*g^2*x^2 + 6*b*f*g*x - 3*b*f^2)*log
(c))*sqrt(h*x))/(h^2*x)

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giac [A]  time = 0.58, size = 649, normalized size = 0.68 \[ \frac {\frac {6 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} + 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-3\right )}}{d h^{2}} + \frac {6 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} - 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-3\right )}}{d h^{2}} + \frac {3 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} e^{\left (-3\right )} \log \left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d h^{2}} - \frac {3 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} e^{\left (-3\right )} \log \left (-\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d h^{2}} + \frac {2 \, {\left (3 \, b g^{2} h^{2} p x^{2} \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b g^{2} h^{2} p x^{2} \log \left (h^{2}\right ) - 4 \, b g^{2} h^{2} p x^{2} + 18 \, b f g h^{2} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 18 \, b f g h^{2} p x \log \left (h^{2}\right ) + 3 \, b g^{2} h^{2} x^{2} \log \relax (c) - 72 \, b f g h^{2} p x + 3 \, a g^{2} h^{2} x^{2} - 9 \, b f^{2} h^{2} p \log \left (h^{2} x^{2} e + d h^{2}\right ) + 9 \, b f^{2} h^{2} p \log \left (h^{2}\right ) + 18 \, b f g h^{2} x \log \relax (c) + 18 \, a f g h^{2} x - 9 \, b f^{2} h^{2} \log \relax (c) - 9 \, a f^{2} h^{2}\right )}}{\sqrt {h x} h^{2}}}{9 \, h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(6*(6*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) + sqrt(2)*(d*h^2)^(3/4)*b*d*g^2*p*e^(9/4) + 3*sqrt(2)*(d*
h^2)^(3/4)*b*f^2*p*e^(13/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(-1/4) + 2*sqrt(h*x))*e^(1/4)/(d*h^2)
^(1/4))*e^(-3)/(d*h^2) + 6*(6*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) + sqrt(2)*(d*h^2)^(3/4)*b*d*g^2*p*e^(
9/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(-1/4) - 2*sqrt(
h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-3)/(d*h^2) + 3*(6*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) - sqrt(2)*(d*h^2
)^(3/4)*b*d*g^2*p*e^(9/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*e^(-3)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*
x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d*h^2) - 3*(6*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) - sqrt(2)*
(d*h^2)^(3/4)*b*d*g^2*p*e^(9/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*e^(-3)*log(-sqrt(2)*(d*h^2)^(1/4)*
sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d*h^2) + 2*(3*b*g^2*h^2*p*x^2*log(h^2*x^2*e + d*h^2) - 3*b*g
^2*h^2*p*x^2*log(h^2) - 4*b*g^2*h^2*p*x^2 + 18*b*f*g*h^2*p*x*log(h^2*x^2*e + d*h^2) - 18*b*f*g*h^2*p*x*log(h^2
) + 3*b*g^2*h^2*x^2*log(c) - 72*b*f*g*h^2*p*x + 3*a*g^2*h^2*x^2 - 9*b*f^2*h^2*p*log(h^2*x^2*e + d*h^2) + 9*b*f
^2*h^2*p*log(h^2) + 18*b*f*g*h^2*x*log(c) + 18*a*f*g*h^2*x - 9*b*f^2*h^2*log(c) - 9*a*f^2*h^2)/(sqrt(h*x)*h^2)
)/h

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maple [F]  time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{2} \left (b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )+a \right )}{\left (h x \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.15, size = 844, normalized size = 0.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*g^2*x^3*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + b*e*f^2*p*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4
)*e^(1/4) + 2*sqrt(h*x)*sqrt(e))/sqrt(sqrt(d)*sqrt(e)*h))/(sqrt(sqrt(d)*sqrt(e)*h)*sqrt(e)) + 2*sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/sqrt(sqrt(d)*sqrt(e)*h))/(sqrt(sqrt(d)*sqr
t(e)*h)*sqrt(e)) - sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/
4)*e^(3/4)) + sqrt(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^
(3/4)))/h + 2/3*a*g^2*x^3/(h*x)^(3/2) + 4*b*f*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(3/2) + 4*a*f*g*x^2/(h*x)^(3/2)
- 2*b*f^2*log((e*x^2 + d)^p*c)/(sqrt(h*x)*h) - 2*(8*sqrt(h*x)*h^2/e - (sqrt(2)*h^4*log(sqrt(e)*h*x + sqrt(2)*
(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) - sqrt(2)*h^4*log(sqrt(e)*h*x - sqrt(2)*(
d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) + 2*sqrt(2)*h^3*arctan(1/2*sqrt(2)*(sqrt(2
)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e))/sqrt(sqrt(d)*sqrt(e)*h))/(sqrt(sqrt(d)*sqrt(e)*h)*sqrt(d)) + 2*
sqrt(2)*h^3*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/sqrt(sqrt(d)*sqrt(e)*h))
/(sqrt(sqrt(d)*sqrt(e)*h)*sqrt(d)))*d/e)*b*e*f*g*p/h^4 - 2*a*f^2/(sqrt(h*x)*h) + 1/9*(3*d*h^4*(2*sqrt(2)*arcta
n(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e))/sqrt(sqrt(d)*sqrt(e)*h))/(sqrt(sqrt(d)*sqr
t(e)*h)*sqrt(e)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/sqrt(sq
rt(d)*sqrt(e)*h))/(sqrt(sqrt(d)*sqrt(e)*h)*sqrt(e)) - sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x
)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) + sqrt(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(
1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)))/e - 8*(h*x)^(3/2)*h^2/e)*b*e*g^2*p/h^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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