3.615 \(\int \frac{(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{9/2}} \, dx\)

Optimal. Leaf size=968 \[ \text{result too large to display} \]

[Out]

(-8*b*e*f^2*p)/(21*d*h^3*(h*x)^(3/2)) - (16*b*e*f*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan
[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1
- (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 -
(Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 + (
Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1 + (Sq
rt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 + (Sqrt
[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*
x)^(7/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) - (2*g^2*(a + b*Log[c*(d + e*x^2)^p]))/(3
*h^3*(h*x)^(3/2)) + (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)
*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2
]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqr
t[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h
] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*f*g*p*L
og[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) + (Sqrt[2]*b*
e^(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*d^(3/4)*h^(9/2)
)

________________________________________________________________________________________

Rubi [A]  time = 1.23186, antiderivative size = 968, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {2467, 2476, 2455, 325, 211, 1165, 628, 1162, 617, 204, 297} \[ \frac{2 \sqrt{2} b e^{7/4} p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{7/4} p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac{2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{7 h (h x)^{7/2}}+\frac{\sqrt{2} b e^{7/4} 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}{7 d^{7/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} 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}{7 d^{7/4} h^{9/2}}-\frac{8 b e p f^2}{21 d h^3 (h x)^{3/2}}+\frac{4 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right ) f}{5 d^{5/4} h^{9/2}}-\frac{4 \sqrt{2} b e^{5/4} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right ) f}{5 d^{5/4} h^{9/2}}-\frac{4 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{5 h^2 (h x)^{5/2}}-\frac{2 \sqrt{2} b e^{5/4} 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}{5 d^{5/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} 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}{5 d^{5/4} h^{9/2}}-\frac{16 b e g p f}{5 d h^4 \sqrt{h x}}-\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]

[Out]

(-8*b*e*f^2*p)/(21*d*h^3*(h*x)^(3/2)) - (16*b*e*f*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan
[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1
- (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 -
(Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 + (
Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1 + (Sq
rt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 + (Sqrt
[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*
x)^(7/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) - (2*g^2*(a + b*Log[c*(d + e*x^2)^p]))/(3
*h^3*(h*x)^(3/2)) + (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)
*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2
]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqr
t[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h
] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*f*g*p*L
og[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) + (Sqrt[2]*b*
e^(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*d^(3/4)*h^(9/2)
)

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 325

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

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)^{9/2}} \, 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^8} \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{f^2 \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{x^8}+\frac{2 f g \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{h x^6}+\frac{g^2 \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{h^2 x^4}\right ) \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{\left (2 g^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^4} \, dx,x,\sqrt{h x}\right )}{h^3}+\frac{(4 f g) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^6} \, 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^8} \, dx,x,\sqrt{h x}\right )}{h}\\ &=-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac{\left (8 b e g^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 h^5}+\frac{(16 b e f g p) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{5 h^4}+\frac{\left (8 b e f^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{7 h^3}\\ &=-\frac{8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac{16 b e f g p}{5 d h^4 \sqrt{h x}}-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac{\left (16 b e^2 f g p\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^6}+\frac{\left (4 b e 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{d} h^6}+\frac{\left (4 b e 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{d} h^6}-\frac{\left (8 b e^2 f^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{7 d h^5}\\ &=-\frac{8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac{16 b e f g p}{5 d h^4 \sqrt{h x}}-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac{\left (4 b e^2 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 )}{7 d^{3/2} h^6}-\frac{\left (4 b e^2 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 )}{7 d^{3/2} h^6}+\frac{\left (8 b e^{3/2} 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 )}{5 d h^6}-\frac{\left (8 b e^{3/2} 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 )}{5 d h^6}-\frac{\left (\sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\left (\sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{\left (2 b \sqrt{e} 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 \sqrt{d} h^4}+\frac{\left (2 b \sqrt{e} 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 \sqrt{d} h^4}\\ &=-\frac{8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac{16 b e f g p}{5 d h^4 \sqrt{h x}}-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{\left (\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac{\left (\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac{\left (2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\left (2 b e^{3/2} 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 )}{7 d^{3/2} h^4}-\frac{\left (2 b e^{3/2} 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 )}{7 d^{3/2} h^4}-\frac{(4 b e 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 )}{5 d h^4}-\frac{(4 b e 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 )}{5 d h^4}\\ &=-\frac{8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac{16 b e f g p}{5 d h^4 \sqrt{h x}}-\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac{\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\left (2 \sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac{\left (2 \sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac{\left (4 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac{\left (4 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}\\ &=-\frac{8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac{16 b e f g p}{5 d h^4 \sqrt{h x}}+\frac{2 \sqrt{2} b e^{7/4} f^2 p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}+\frac{4 \sqrt{2} b e^{5/4} f g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{7/4} f^2 p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{7 d^{7/4} h^{9/2}}-\frac{4 \sqrt{2} b e^{5/4} f g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac{4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac{2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac{\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac{2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}-\frac{\sqrt{2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac{2 \sqrt{2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac{\sqrt{2} b e^{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 d^{3/4} h^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.262752, size = 294, normalized size = 0.3 \[ \frac{x \left (-30 d f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-84 d f g x \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-70 d g^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-35 \sqrt{2} b \sqrt [4]{d} e^{3/4} g^2 p x^{7/2} \left (\log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-\log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )\right )-40 b e f^2 p x^2 \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{e x^2}{d}\right )-336 b e f g p x^3 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{e x^2}{d}\right )\right )}{105 d (h x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(9/2),x]

[Out]

(x*(-40*b*e*f^2*p*x^2*Hypergeometric2F1[-3/4, 1, 1/4, -((e*x^2)/d)] - 336*b*e*f*g*p*x^3*Hypergeometric2F1[-1/4
, 1, 3/4, -((e*x^2)/d)] - 35*Sqrt[2]*b*d^(1/4)*e^(3/4)*g^2*p*x^(7/2)*(2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d
^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sq
rt[e]*x] - Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]) - 30*d*f^2*(a + b*Log[c*(d + e*x^2)^p])
- 84*d*f*g*x*(a + b*Log[c*(d + e*x^2)^p]) - 70*d*g^2*x^2*(a + b*Log[c*(d + e*x^2)^p])))/(105*d*(h*x)^(9/2))

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Maple [F]  time = 1.354, 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 ) \left ( hx \right ) ^{-{\frac{9}{2}}}}\, 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)^(9/2),x)

[Out]

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(9/2),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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.56233, size = 5276, normalized size = 5.45 \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)^(9/2),x, algorithm="fricas")

[Out]

-2/105*(d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^
4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*
d*e^2*f*g^3)*p^2)/(d^3*h^9))*log(16*(50625*b^3*e^6*f^8 - 472500*b^3*d*e^5*f^6*g^2 - 1457946*b^3*d^2*e^4*f^4*g^
4 - 2572500*b^3*d^3*e^3*f^2*g^6 + 1500625*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^3 + 16*(42*d^6*f*g*h^14*sqrt(-(50625*b^
4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^
4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 5*(675*b^2*d^2*e^4*f^6 - 10017*b^2*d^3*e^3*f^4*g^2 + 23373*b^2*d^4*e^2*f^2*g^
4 - 8575*b^2*d^5*e*g^6)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846
*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3
*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))) - d*h^5*x^4*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4
*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7
*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))*log(16*(50625*b^3*e^6*f^8 - 472500*b^3*d*e
^5*f^6*g^2 - 1457946*b^3*d^2*e^4*f^4*g^4 - 2572500*b^3*d^3*e^3*f^2*g^6 + 1500625*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^
3 - 16*(42*d^6*f*g*h^14*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6
894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) + 5*(675*b^2*d^2*e^4*f^6 - 10017*b^2*d^3
*e^3*f^4*g^2 + 23373*b^2*d^4*e^2*f^2*g^4 - 8575*b^2*d^5*e*g^6)*h^5*p^2)*sqrt(-(d^3*h^9*sqrt(-(50625*b^4*e^7*f^
8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^
3*g^8)*p^4/(d^7*h^18)) + 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))) - d*h^5*x^4*sqrt((d^3*h^9*
sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g
^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) - 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))*log(
16*(50625*b^3*e^6*f^8 - 472500*b^3*d*e^5*f^6*g^2 - 1457946*b^3*d^2*e^4*f^4*g^4 - 2572500*b^3*d^3*e^3*f^2*g^6 +
1500625*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^3 + 16*(42*d^6*f*g*h^14*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6
*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) -
5*(675*b^2*d^2*e^4*f^6 - 10017*b^2*d^3*e^3*f^4*g^2 + 23373*b^2*d^4*e^2*f^2*g^4 - 8575*b^2*d^5*e*g^6)*h^5*p^2)*
sqrt((d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4
*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) - 420*(3*b^2*e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(
d^3*h^9))) + d*h^5*x^4*sqrt((d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^
5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) - 420*(3*b^2*e^3*f^3*g - 7*
b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))*log(16*(50625*b^3*e^6*f^8 - 472500*b^3*d*e^5*f^6*g^2 - 1457946*b^3*d^2*e^4*f^
4*g^4 - 2572500*b^3*d^3*e^3*f^2*g^6 + 1500625*b^3*d^4*e^2*g^8)*sqrt(h*x)*p^3 - 16*(42*d^6*f*g*h^14*sqrt(-(5062
5*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 150062
5*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) - 5*(675*b^2*d^2*e^4*f^6 - 10017*b^2*d^3*e^3*f^4*g^2 + 23373*b^2*d^4*e^2*f^
2*g^4 - 8575*b^2*d^5*e*g^6)*h^5*p^2)*sqrt((d^3*h^9*sqrt(-(50625*b^4*e^7*f^8 - 1266300*b^4*d*e^6*f^6*g^2 + 8469
846*b^4*d^2*e^5*f^4*g^4 - 6894300*b^4*d^3*e^4*f^2*g^6 + 1500625*b^4*d^4*e^3*g^8)*p^4/(d^7*h^18)) - 420*(3*b^2*
e^3*f^3*g - 7*b^2*d*e^2*f*g^3)*p^2)/(d^3*h^9))) + (168*b*e*f*g*p*x^3 + 42*a*d*f*g*x + 15*a*d*f^2 + 5*(4*b*e*f^
2*p + 7*a*d*g^2)*x^2 + (35*b*d*g^2*p*x^2 + 42*b*d*f*g*p*x + 15*b*d*f^2*p)*log(e*x^2 + d) + (35*b*d*g^2*x^2 + 4
2*b*d*f*g*x + 15*b*d*f^2)*log(c))*sqrt(h*x))/(d*h^5*x^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.58614, size = 911, normalized size = 0.94 \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)^(9/2),x, algorithm="giac")

[Out]

1/105*(2*(35*sqrt(2)*(d*h^2)^(1/4)*b*d*g^2*h*p*e^(7/4) - 15*sqrt(2)*(d*h^2)^(1/4)*b*f^2*h*p*e^(11/4) - 42*sqrt
(2)*(d*h^2)^(3/4)*b*f*g*p*e^(9/4))*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^(-1)/(d^2*h^2) + 2*(35*sqrt(2)*(d*h^2)^(1/4)*b*d*g^2*h*p*e^(7/4) - 15*sqrt(2)*(d*h^2)^(1/4)*b*
f^2*h*p*e^(11/4) - 42*sqrt(2)*(d*h^2)^(3/4)*b*f*g*p*e^(9/4))*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^(-1)/(d^2*h^2) + (35*sqrt(2)*(d*h^2)^(1/4)*b*d*g^2*h*p*e^(7/4) - 15
*sqrt(2)*(d*h^2)^(1/4)*b*f^2*h*p*e^(11/4) + 42*sqrt(2)*(d*h^2)^(3/4)*b*f*g*p*e^(9/4))*e^(-1)*log(sqrt(2)*(d*h^
2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d^2*h^2) - (35*sqrt(2)*(d*h^2)^(1/4)*b*d*g^2*h*p*e^
(7/4) - 15*sqrt(2)*(d*h^2)^(1/4)*b*f^2*h*p*e^(11/4) + 42*sqrt(2)*(d*h^2)^(3/4)*b*f*g*p*e^(9/4))*e^(-1)*log(-sq
rt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d^2*h^2))/h^4 - 2/105*(168*b*f*g*h^4*p*x
^3*e + 35*b*d*g^2*h^4*p*x^2*log(h^2*x^2*e + d*h^2) - 35*b*d*g^2*h^4*p*x^2*log(h^2) + 20*b*f^2*h^4*p*x^2*e + 42
*b*d*f*g*h^4*p*x*log(h^2*x^2*e + d*h^2) - 42*b*d*f*g*h^4*p*x*log(h^2) + 35*b*d*g^2*h^4*x^2*log(c) + 35*a*d*g^2
*h^4*x^2 + 15*b*d*f^2*h^4*p*log(h^2*x^2*e + d*h^2) - 15*b*d*f^2*h^4*p*log(h^2) + 42*b*d*f*g*h^4*x*log(c) + 42*
a*d*f*g*h^4*x + 15*b*d*f^2*h^4*log(c) + 15*a*d*f^2*h^4)/(sqrt(h*x)*d*h^8*x^3)