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

Optimal. Leaf size=1002 \[ \text{result too large to display} \]

[Out]

(2*a*f^2*Sqrt[h*x])/h - (8*b*f^2*p*Sqrt[h*x])/h + (8*b*d*g^2*p*Sqrt[h*x])/(5*e*h) - (16*b*f*g*p*(h*x)^(3/2))/(
9*h^2) - (8*b*g^2*p*(h*x)^(5/2))/(25*h^3) - (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/
(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) - (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(
1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/
4)*Sqrt[h])])/(5*e^(5/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)
*Sqrt[h])])/(e^(1/4)*Sqrt[h]) + (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqr
t[h])])/(3*e^(3/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[
h])])/(5*e^(5/4)*Sqrt[h]) + (2*b*f^2*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h + (4*f*g*(h*x)^(3/2)*(a + b*Log[c*(d +
e*x^2)^p]))/(3*h^2) + (2*g^2*(h*x)^(5/2)*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^3) - (Sqrt[2]*b*d^(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]])/(e^(1/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(
3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) +
(Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(
5/4)*Sqrt[h]) + (Sqrt[2]*b*d^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqr
t[h*x]])/(e^(1/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1
/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) - (Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x
+ Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(5/4)*Sqrt[h])

________________________________________________________________________________________

Rubi [A]  time = 1.30943, antiderivative size = 1002, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {2467, 2471, 2448, 321, 211, 1165, 628, 1162, 617, 204, 2455, 297, 302} \[ -\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac{2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{5/2}}{5 h^3}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}+\frac{4 f g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{3/2}}{3 h^2}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}+\frac{2 b f^2 \log \left (c \left (e x^2+d\right )^p\right ) \sqrt{h x}}{h}+\frac{2 a f^2 \sqrt{h x}}{h}-\frac{2 \sqrt{2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{e} \sqrt{h}}+\frac{2 \sqrt{2} b d^{5/4} g^2 p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 e^{5/4} \sqrt{h}}-\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 e^{3/4} \sqrt{h}}+\frac{2 \sqrt{2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{e} \sqrt{h}}-\frac{2 \sqrt{2} b d^{5/4} g^2 p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{5 e^{5/4} \sqrt{h}}+\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{3 e^{3/4} \sqrt{h}}-\frac{\sqrt{2} b \sqrt [4]{d} f^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 )}{\sqrt [4]{e} \sqrt{h}}+\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{2 \sqrt{2} b d^{3/4} f 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\sqrt{2} b \sqrt [4]{d} f^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 )}{\sqrt [4]{e} \sqrt{h}}-\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}-\frac{2 \sqrt{2} b d^{3/4} f 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 )}{3 e^{3/4} \sqrt{h}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*f^2*Sqrt[h*x])/h - (8*b*f^2*p*Sqrt[h*x])/h + (8*b*d*g^2*p*Sqrt[h*x])/(5*e*h) - (16*b*f*g*p*(h*x)^(3/2))/(
9*h^2) - (8*b*g^2*p*(h*x)^(5/2))/(25*h^3) - (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/
(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) - (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(
1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/
4)*Sqrt[h])])/(5*e^(5/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)
*Sqrt[h])])/(e^(1/4)*Sqrt[h]) + (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqr
t[h])])/(3*e^(3/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[
h])])/(5*e^(5/4)*Sqrt[h]) + (2*b*f^2*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h + (4*f*g*(h*x)^(3/2)*(a + b*Log[c*(d +
e*x^2)^p]))/(3*h^2) + (2*g^2*(h*x)^(5/2)*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^3) - (Sqrt[2]*b*d^(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]])/(e^(1/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(
3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) +
(Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(
5/4)*Sqrt[h]) + (Sqrt[2]*b*d^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqr
t[h*x]])/(e^(1/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1
/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) - (Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x
+ Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(5/4)*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 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

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 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 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 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 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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 )}{\sqrt{h x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \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 ) \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (f^2 \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )+\frac{2 f g x^2 \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{h}+\frac{g^2 x^4 \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^4 \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 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^2}+\frac{\left (2 f^2\right ) \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}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac{\left (2 b f^2\right ) \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}-\frac{\left (8 b e g^2 p\right ) \operatorname{Subst}\left (\int \frac{x^8}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 h^5}-\frac{(16 b e f g p) \operatorname{Subst}\left (\int \frac{x^6}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 h^4}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac{\left (8 b e g^2 p\right ) \operatorname{Subst}\left (\int \left (-\frac{d h^4}{e^2}+\frac{h^2 x^4}{e}+\frac{d^2 h^4}{e^2 \left (d+\frac{e x^4}{h^2}\right )}\right ) \, dx,x,\sqrt{h x}\right )}{5 h^5}-\frac{\left (8 b e f^2 p\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{h^3}+\frac{(16 b d f g p) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 h^2}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}-\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac{(8 b d f g p) \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^2}+\frac{(8 b d f g p) \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^2}+\frac{\left (8 b d f^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{h}-\frac{\left (8 b d^2 g^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 e h}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}-\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac{(4 b d f g 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 )}{3 e}+\frac{(4 b d f g 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 )}{3 e}+\frac{\left (4 b \sqrt{d} 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^2}+\frac{\left (4 b \sqrt{d} 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^2}-\frac{\left (4 b d^{3/2} 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 )}{5 e h^2}-\frac{\left (4 b d^{3/2} 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 )}{5 e h^2}+\frac{\left (2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\left (2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}-\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}-\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\left (2 b \sqrt{d} 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 )}{\sqrt{e}}+\frac{\left (2 b \sqrt{d} 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 )}{\sqrt{e}}-\frac{\left (2 b d^{3/2} 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 )}{5 e^{3/2}}-\frac{\left (2 b d^{3/2} 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 )}{5 e^{3/2}}-\frac{\left (\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}-\frac{\left (\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}+\frac{\left (4 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}-\frac{\left (4 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\left (\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{\left (\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}-\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 e^{3/4} \sqrt{h}}+\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 e^{3/4} \sqrt{h}}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac{\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}+\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}-\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}-\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{\left (2 \sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}-\frac{\left (2 \sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}-\frac{\left (2 \sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{\left (2 \sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}\\ &=\frac{2 a f^2 \sqrt{h x}}{h}-\frac{8 b f^2 p \sqrt{h x}}{h}+\frac{8 b d g^2 p \sqrt{h x}}{5 e h}-\frac{16 b f g p (h x)^{3/2}}{9 h^2}-\frac{8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac{2 \sqrt{2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{e} \sqrt{h}}-\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 e^{3/4} \sqrt{h}}+\frac{2 \sqrt{2} b d^{5/4} g^2 p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 e^{5/4} \sqrt{h}}+\frac{2 \sqrt{2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{e} \sqrt{h}}+\frac{4 \sqrt{2} b d^{3/4} f g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 e^{3/4} \sqrt{h}}-\frac{2 \sqrt{2} b d^{5/4} g^2 p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 e^{5/4} \sqrt{h}}+\frac{2 b f^2 \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac{4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac{2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac{\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}+\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}+\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}+\frac{\sqrt{2} b \sqrt [4]{d} 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]{e} \sqrt{h}}-\frac{2 \sqrt{2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt{h}}-\frac{\sqrt{2} b d^{5/4} 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 )}{5 e^{5/4} \sqrt{h}}\\ \end{align*}

Mathematica [A]  time = 1.33385, size = 588, normalized size = 0.59 \[ \frac{2 \sqrt{x} \left (\frac{2}{3} f g x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+\frac{1}{5} g^2 x^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+a f^2 \sqrt{x}+b f^2 \sqrt{x} \log \left (c \left (d+e x^2\right )^p\right )-\frac{b g^2 p \left (-5 \sqrt{2} d^{5/4} \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+5 \sqrt{2} d^{5/4} \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-10 \sqrt{2} d^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )+10 \sqrt{2} d^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )-40 d \sqrt [4]{e} \sqrt{x}+8 e^{5/4} x^{5/2}\right )}{50 e^{5/4}}-\frac{4 b f g p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}-\frac{b f^2 p \left (\sqrt{2} \sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-\sqrt{2} \sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )-2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )+8 \sqrt [4]{e} \sqrt{x}\right )}{2 \sqrt [4]{e}}\right )}{\sqrt{h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(a*f^2*Sqrt[x] - (4*b*f*g*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)) - (b*f^2*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)*Lo
g[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(2*e^(1/4)) - (b*g^2*p*(-40*d*e^(1/4)*Sqrt[x] + 8*e
^(5/4)*x^(5/2) - 10*Sqrt[2]*d^(5/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + 10*Sqrt[2]*d^(5/4)*ArcTan[
1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 5*Sqrt[2]*d^(5/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqr
t[e]*x] + 5*Sqrt[2]*d^(5/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(50*e^(5/4)) + b*f^2*
Sqrt[x]*Log[c*(d + e*x^2)^p] + (2*f*g*x^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/3 + (g^2*x^(5/2)*(a + b*Log[c*(d +
e*x^2)^p]))/5))/Sqrt[h*x]

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Maple [F]  time = 1.281, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ){\frac{1}{\sqrt{hx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.36029, size = 4782, normalized size = 4.77 \begin{align*} \text{result too large to display} \end{align*}

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)^(1/2),x, algorithm="fricas")

[Out]

2/225*(15*e*h*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4
- 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))
*log(16*(50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150*b^3*d^2*e^2*f^4*g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*
b^3*d^4*g^8)*sqrt(h*x)*p^3 + 16*(10*e^4*f*g*h^2*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150
*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) + 3*(1125*b^2*e^4*f^6 - 1175*b^
2*d*e^3*f^4*g^2 + 235*b^2*d^2*e^2*f^2*g^4 - 9*b^2*d^3*e*g^6)*h*p^2)*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 -
85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)
) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))) - 15*e*h*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 855
00*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) +
60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))*log(16*(50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150
*b^3*d^2*e^2*f^4*g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 - 16*(10*e^4*f*g*h^2*sqrt(-(5062
5*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*
g^8)*p^4/(e^5*h^2)) + 3*(1125*b^2*e^4*f^6 - 1175*b^2*d*e^3*f^4*g^2 + 235*b^2*d^2*e^2*f^2*g^4 - 9*b^2*d^3*e*g^6
)*h*p^2)*sqrt(-(e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 342
0*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) + 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))) - 1
5*e*h*sqrt((e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^
4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) - 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))*log(16*(
50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150*b^3*d^2*e^2*f^4*g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g
^8)*sqrt(h*x)*p^3 + 16*(10*e^4*f*g*h^2*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*
e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) - 3*(1125*b^2*e^4*f^6 - 1175*b^2*d*e^3*f
^4*g^2 + 235*b^2*d^2*e^2*f^2*g^4 - 9*b^2*d^3*e*g^6)*h*p^2)*sqrt((e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*
d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) - 60*(5*
b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))) + 15*e*h*sqrt((e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*
e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) - 60*(5*b^2*
d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))*log(16*(50625*b^3*e^4*f^8 - 40500*b^3*d*e^3*f^6*g^2 + 2150*b^3*d^2*e^
2*f^4*g^4 - 1620*b^3*d^3*e*f^2*g^6 + 81*b^3*d^4*g^8)*sqrt(h*x)*p^3 - 16*(10*e^4*f*g*h^2*sqrt(-(50625*b^4*d*e^4
*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e
^5*h^2)) - 3*(1125*b^2*e^4*f^6 - 1175*b^2*d*e^3*f^4*g^2 + 235*b^2*d^2*e^2*f^2*g^4 - 9*b^2*d^3*e*g^6)*h*p^2)*sq
rt((e^2*h*sqrt(-(50625*b^4*d*e^4*f^8 - 85500*b^4*d^2*e^3*f^6*g^2 + 40150*b^4*d^3*e^2*f^4*g^4 - 3420*b^4*d^4*e*
f^2*g^6 + 81*b^4*d^5*g^8)*p^4/(e^5*h^2)) - 60*(5*b^2*d*e*f^3*g - b^2*d^2*f*g^3)*p^2)/(e^2*h))) + (225*a*e*f^2
- 9*(4*b*e*g^2*p - 5*a*e*g^2)*x^2 - 180*(5*b*e*f^2 - b*d*g^2)*p - 50*(4*b*e*f*g*p - 3*a*e*f*g)*x + 15*(3*b*e*g
^2*p*x^2 + 10*b*e*f*g*p*x + 15*b*e*f^2*p)*log(e*x^2 + d) + 15*(3*b*e*g^2*x^2 + 10*b*e*f*g*x + 15*b*e*f^2)*log(
c))*sqrt(h*x))/(e*h)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.38448, size = 1107, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

1/225*(90*sqrt(h*x)*b*g^2*x^2*log(c) + 90*sqrt(h*x)*a*g^2*x^2 + 300*sqrt(h*x)*b*f*g*x*log(c) + 225*((2*sqrt(2)
*(d*h^2)^(1/4)*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^(-5/
4) + 2*sqrt(2)*(d*h^2)^(1/4)*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^(-5/4) + sqrt(2)*(d*h^2)^(1/4)*e^(-5/4)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d
*h^2)*e^(-1/2)) - sqrt(2)*(d*h^2)^(1/4)*e^(-5/4)*log(-sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*
h^2)*e^(-1/2)) - 8*sqrt(h*x)*e^(-1))*e + 2*sqrt(h*x)*log(x^2*e + d))*b*f^2*p + 9*(10*sqrt(h*x)*x^2*log(x^2*e +
d) - (10*sqrt(2)*(d*h^2)^(1/4)*d*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^(-9/4) + 10*sqrt(2)*(d*h^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(-1/4) - 2*sqr
t(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-9/4) + 5*sqrt(2)*(d*h^2)^(1/4)*d*e^(-9/4)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*
x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2)) - 5*sqrt(2)*(d*h^2)^(1/4)*d*e^(-9/4)*log(-sqrt(2)*(d*h^2)^(1/4)*sqrt
(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2)) + 8*(sqrt(h*x)*h^10*x^2*e^4 - 5*sqrt(h*x)*d*h^10*e^3)*e^(-5)/h^10
)*e)*b*g^2*p + 300*sqrt(h*x)*a*f*g*x + 450*sqrt(h*x)*b*f^2*log(c) + 50*(6*sqrt(h*x)*h*x*log(x^2*e + d) - (8*sq
rt(h*x)*h*x*e^(-1) - 6*sqrt(2)*(d*h^2)^(3/4)*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^(-7/4) - 6*sqrt(2)*(d*h^2)^(3/4)*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^(-7/4) + 3*sqrt(2)*(d*h^2)^(3/4)*e^(-7/4)*log(sqrt(2)*(d*h^2)^(1/4)*s
qrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2)) - 3*sqrt(2)*(d*h^2)^(3/4)*e^(-7/4)*log(-sqrt(2)*(d*h^2)^(1/4)*
sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2)))*e)*b*f*g*p/h + 450*sqrt(h*x)*a*f^2)/h