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

Optimal. Leaf size=603 \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt{h x}}+\frac{2 a g \sqrt{h x}}{h^2}+\frac{2 b g \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}+\frac{\sqrt{2} b \sqrt [4]{e} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac{2 \sqrt{2} b \sqrt [4]{e} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{d} h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{d} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{e} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{d} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{e} h^{3/2}}-\frac{2 \sqrt{2} b \sqrt [4]{d} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{d} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{e} h^{3/2}}-\frac{8 b g p \sqrt{h x}}{h^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.794842, antiderivative size = 603, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2467, 2476, 2448, 321, 211, 1165, 628, 1162, 617, 204, 2455, 297} \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt{h x}}+\frac{2 a g \sqrt{h x}}{h^2}+\frac{2 b g \sqrt{h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}+\frac{\sqrt{2} b \sqrt [4]{e} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{e} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{d} h^{3/2}}-\frac{2 \sqrt{2} b \sqrt [4]{e} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{e} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{d} h^{3/2}}-\frac{\sqrt{2} b \sqrt [4]{d} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{e} h^{3/2}}+\frac{\sqrt{2} b \sqrt [4]{d} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{\sqrt [4]{e} h^{3/2}}-\frac{2 \sqrt{2} b \sqrt [4]{d} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac{2 \sqrt{2} b \sqrt [4]{d} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{\sqrt [4]{e} h^{3/2}}-\frac{8 b g p \sqrt{h x}}{h^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2467

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r
_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + (g*x^k)/h)^r*(a + b*Lo
g[c*(d + (e*x^(k*n))/h^n)^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && Fract
ionQ[m] && IntegerQ[n] && IntegerQ[r]

Rule 2476

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

Rule 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]]))

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

Mathematica [A]  time = 0.477263, size = 316, normalized size = 0.52 \[ \frac{2 x^{3/2} \left (-\frac{f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt{x}}+a g \sqrt{x}+b g \sqrt{x} \log \left (c \left (d+e x^2\right )^p\right )+\frac{2 b \sqrt [4]{e} f p \left (\tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac{d \sqrt [4]{e} \sqrt{x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac{b g p \left (\sqrt{2} \sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-\sqrt{2} \sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )-2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )+8 \sqrt [4]{e} \sqrt{x}\right )}{2 \sqrt [4]{e}}\right )}{(h x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.149, 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{3}{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)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.98322, size = 2245, normalized size = 3.72 \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)^(3/2),x, algorithm="fricas")

[Out]

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

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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)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.58716, size = 576, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} + \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f 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^{3}} + \frac{2 \,{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} + \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f 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^{3}} + \frac{{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} - \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f 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^{3}} - \frac{{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} - \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f 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^{3}} + \frac{2 \,{\left (b g h p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - b g h p x \log \left (h^{2}\right ) - 4 \, b g h p x - b f h p \log \left (h^{2} x^{2} e + d h^{2}\right ) + b f h p \log \left (h^{2}\right ) + b g h x \log \left (c\right ) + a g h x - b f h \log \left (c\right ) - a f h\right )}}{\sqrt{h x} h^{2}} \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)^(3/2),x, algorithm="giac")

[Out]

2*(sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) + sqrt(2)*(d*h^2)^(3/4)*b*f*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^3) + 2*(sqrt(2)*(d*h^2)^(1/4)*b*d*g*h
*p*e^(7/4) + sqrt(2)*(d*h^2)^(3/4)*b*f*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^3) + (sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) - sqrt(2)*(d*h^2)^(3/4
)*b*f*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^3) - (
sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) - sqrt(2)*(d*h^2)^(3/4)*b*f*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^3) + 2*(b*g*h*p*x*log(h^2*x^2*e + d*h^2) - b*g*h*p*x*
log(h^2) - 4*b*g*h*p*x - b*f*h*p*log(h^2*x^2*e + d*h^2) + b*f*h*p*log(h^2) + b*g*h*x*log(c) + a*g*h*x - b*f*h*
log(c) - a*f*h)/(sqrt(h*x)*h^2)