3.608 \(\int \frac{(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{5/2}} \, dx\)

Optimal. Leaf size=588 \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt{h x}}-\frac{\sqrt{2} b e^{3/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{5/2}}+\frac{\sqrt{2} b e^{3/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{5/2}}-\frac{2 \sqrt{2} b e^{3/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{5/2}}+\frac{2 \sqrt{2} b e^{3/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{3 d^{3/4} h^{5/2}}+\frac{\sqrt{2} b \sqrt [4]{e} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{5/2}}-\frac{\sqrt{2} b \sqrt [4]{e} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{5/2}}-\frac{2 \sqrt{2} b \sqrt [4]{e} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{d} h^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.74087, antiderivative size = 588, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2467, 2476, 2455, 211, 1165, 628, 1162, 617, 204, 297} \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt{h x}}-\frac{\sqrt{2} b e^{3/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{5/2}}+\frac{\sqrt{2} b e^{3/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{5/2}}-\frac{2 \sqrt{2} b e^{3/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{5/2}}+\frac{2 \sqrt{2} b e^{3/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{3 d^{3/4} h^{5/2}}+\frac{\sqrt{2} b \sqrt [4]{e} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{5/2}}-\frac{\sqrt{2} b \sqrt [4]{e} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{5/2}}-\frac{2 \sqrt{2} b \sqrt [4]{e} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{d} h^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.433418, size = 271, normalized size = 0.46 \[ \frac{2 x^{5/2} \left (-\frac{f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac{g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt{x}}-\frac{b e^{3/4} f p \left (\log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-\log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )\right )}{3 \sqrt{2} d^{3/4}}+\frac{2 b \sqrt [4]{e} g p \left (\tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac{d \sqrt [4]{e} \sqrt{x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}\right )}{(h x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]

[Out]

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

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Maple [F]  time = 1.139, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) \left ( hx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(5/2),x)

[Out]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(5/2),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)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.95346, size = 2557, normalized size = 4.35 \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)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4
/(d^3*h^10)))/(d*h^5))*log(-32*(b^3*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 + 32*(3*d^3*g*h^8*sqrt(-(b^4*e^3
*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) + (b^2*d*e^2*f^3 - 9*b^2*d^2*e*f*g^2)*h^3*p^2)
*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))
/(d*h^5))) - h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g
^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*(b^3*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 - 32*(3*d^3*g*h^8*sqrt(-(
b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) + (b^2*d*e^2*f^3 - 9*b^2*d^2*e*f*g^2)*h
^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 + d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*
h^10)))/(d*h^5))) - h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*
d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*(b^3*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 + 32*(3*d^3*g*h^8*
sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) - (b^2*d*e^2*f^3 - 9*b^2*d^2*e*f
*g^2)*h^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^
4/(d^3*h^10)))/(d*h^5))) + h^3*x^2*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 +
81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)))/(d*h^5))*log(-32*(b^3*e^3*f^4 - 81*b^3*d^2*e*g^4)*sqrt(h*x)*p^3 - 32*(3*d^3
*g*h^8*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*g^4)*p^4/(d^3*h^10)) - (b^2*d*e^2*f^3 - 9*b^2*
d^2*e*f*g^2)*h^3*p^2)*sqrt(-(6*b^2*e*f*g*p^2 - d*h^5*sqrt(-(b^4*e^3*f^4 - 18*b^4*d*e^2*f^2*g^2 + 81*b^4*d^2*e*
g^4)*p^4/(d^3*h^10)))/(d*h^5))) + (3*a*g*x + a*f + (3*b*g*p*x + b*f*p)*log(e*x^2 + d) + (3*b*g*x + b*f)*log(c)
)*sqrt(h*x))/(h^3*x^2)

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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)*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.65191, size = 599, normalized size = 1.02 \begin{align*} \frac{\frac{2 \,{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b f h p e^{\frac{11}{4}} + 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b g p e^{\frac{9}{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 (-2\right )}}{d h^{2}} + \frac{2 \,{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b f h p e^{\frac{11}{4}} + 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b g p e^{\frac{9}{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 (-2\right )}}{d h^{2}} + \frac{{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b f h p e^{\frac{11}{4}} - 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b g p e^{\frac{9}{4}}\right )} e^{\left (-2\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{{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b f h p e^{\frac{11}{4}} - 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b g p e^{\frac{9}{4}}\right )} e^{\left (-2\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}}}{3 \, h^{2}} - \frac{2 \,{\left (3 \, b g h^{2} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b g h^{2} p x \log \left (h^{2}\right ) + b f h^{2} p \log \left (h^{2} x^{2} e + d h^{2}\right ) - b f h^{2} p \log \left (h^{2}\right ) + 3 \, b g h^{2} x \log \left (c\right ) + 3 \, a g h^{2} x + b f h^{2} \log \left (c\right ) + a f h^{2}\right )}}{3 \, \sqrt{h x} h^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(5/2),x, algorithm="giac")

[Out]

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