\(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{\sqrt {h x}} \, dx\) [606]

Optimal result

Integrand size = 29, antiderivative size = 631 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2517, 2521, 2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303} \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 a f \sqrt {h x}}{h}-\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {\sqrt {2} b d^{3/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2} \]

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

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 303

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 327

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 631

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 642

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 1176

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 1179

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 2498

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 2505

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 2517

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 2521

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]))

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.59 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 \sqrt {x} \left (a f \sqrt {x}-4 b f p \sqrt {x}+\frac {1}{3} a g x^{3/2}-\frac {4}{9} b g p x^{3/2}-\frac {\sqrt {2} b \sqrt [4]{d} f p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}-\frac {2 b (-d)^{3/4} g p \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {2 b (-d)^{3/4} g p \text {arctanh}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}-\frac {b \sqrt [4]{d} f p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} f p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+b f \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} b g x^{3/2} \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\sqrt {h x}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1196 vs. \(2 (441) = 882\).

Time = 0.34 (sec) , antiderivative size = 1196, normalized size of antiderivative = 1.90 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2/9*(3*h*sqrt(-(6*b^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*
h^2)))/(e*h))*log(-32*(81*b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 1
8*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)) + 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*
g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) - 3*h*sqrt(-
(6*b^2*d*f*g*p^2 + e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))*lo
g(-32*(81*b^3*e^2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 - 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*
g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)) + 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*g*p^2 + e*h*sqrt
(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) - 3*h*sqrt(-(6*b^2*d*f*g*p^2
- e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))*log(-32*(81*b^3*e^
2*f^4 - b^3*d^2*g^4)*sqrt(h*x)*p^3 + 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^
4)*p^4/(e^3*h^2)) - 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*g*p^2 - e*h*sqrt(-(81*b^4*d*e^2*
f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) + 3*h*sqrt(-(6*b^2*d*f*g*p^2 - e*h*sqrt(-(81
*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))*log(-32*(81*b^3*e^2*f^4 - b^3*d^2*
g^4)*sqrt(h*x)*p^3 - 32*(e^2*g*h^2*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)
) - 3*(9*b^2*e^2*f^3 - b^2*d*e*f*g^2)*h*p^2)*sqrt(-(6*b^2*d*f*g*p^2 - e*h*sqrt(-(81*b^4*d*e^2*f^4 - 18*b^4*d^2
*e*f^2*g^2 + b^4*d^3*g^4)*p^4/(e^3*h^2)))/(e*h))) + (36*b*f*p - 9*a*f + (4*b*g*p - 3*a*g)*x - 3*(b*g*p*x + 3*b
*f*p)*log(e*x^2 + d) - 3*(b*g*x + 3*b*f)*log(c))*sqrt(h*x))/h

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.19 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 \, b g x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{3 \, \sqrt {h x}} + \frac {2 \, a g x^{2}}{3 \, \sqrt {h x}} + \frac {2 \, \sqrt {h x} b f \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{h} - \frac {{\left (\frac {8 \, \sqrt {h x} h^{2}}{e} - \frac {{\left (\frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} - \frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}} e^{\frac {1}{4}}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}} + \frac {\sqrt {2} h^{3} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {d}}\right )} d}{e}\right )} b e f p}{h^{3}} + \frac {2 \, \sqrt {h x} a f}{h} - \frac {{\left (\frac {3 \, d h^{4} {\left (\frac {\sqrt {2} \log \left (\sqrt {e} h x + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {e} h x - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} \sqrt {e}}{\sqrt {2} \sqrt {-\sqrt {d} \sqrt {e} h} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} \sqrt {e}}\right )}{\sqrt {-\sqrt {d} \sqrt {e} h} \sqrt {e}}\right )}}{e} + \frac {8 \, \left (h x\right )^{\frac {3}{2}} h^{2}}{e}\right )} b e g p}{9 \, h^{4}} \]

[In]

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

[Out]

2/3*b*g*x^2*log((e*x^2 + d)^p*c)/sqrt(h*x) + 2/3*a*g*x^2/sqrt(h*x) + 2*sqrt(h*x)*b*f*log((e*x^2 + d)^p*c)/h -
(8*sqrt(h*x)*h^2/e - (sqrt(2)*h^4*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h
^2)^(3/4)*e^(1/4)) - sqrt(2)*h^4*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^
2)^(3/4)*e^(1/4)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqr
t(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqr
t(-sqrt(d)*sqrt(e)*h)*sqrt(d)) + sqrt(2)*h^3*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^
(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*s
qrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(d)))*d/e)*b*e*f*p/h^3 + 2*sqrt(h*x)*a*f/h - 1/9*(3*d*h^4*(sqrt(2)*log(
sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(
e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(-(sqrt(2)*
sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)
*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)) - sqrt(2)*log(-
(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d
)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)))/e + 8
*(h*x)^(3/2)*h^2/e)*b*e*g*p/h^4

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {6 \, \sqrt {h x} b g x \log \left (c\right ) + 9 \, {\left (e {\left (\frac {2 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{e^{2}} + \frac {2 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{e^{2}} + \frac {\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{e^{2}} - \frac {\sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{e^{2}} - \frac {8 \, \sqrt {h x}}{e}\right )} + 2 \, \sqrt {h x} \log \left (e x^{2} + d\right )\right )} b f p + 6 \, \sqrt {h x} a g x + 18 \, \sqrt {h x} b f \log \left (c\right ) + \frac {{\left (6 \, \sqrt {h x} h x \log \left (e x^{2} + d\right ) - {\left (\frac {8 \, \sqrt {h x} h x}{e} - \frac {6 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{e^{4}} - \frac {6 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{e^{4}} + \frac {3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{e^{4}} - \frac {3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{e^{4}}\right )} e\right )} b g p}{h} + 18 \, \sqrt {h x} a f}{9 \, h} \]

[In]

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

[Out]

1/9*(6*sqrt(h*x)*b*g*x*log(c) + 9*(e*(2*sqrt(2)*(d*e^3*h^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4)
+ 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^2 + 2*sqrt(2)*(d*e^3*h^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4)
- 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^2 + sqrt(2)*(d*e^3*h^2)^(1/4)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) +
sqrt(d*h^2/e))/e^2 - sqrt(2)*(d*e^3*h^2)^(1/4)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/e
^2 - 8*sqrt(h*x)/e) + 2*sqrt(h*x)*log(e*x^2 + d))*b*f*p + 6*sqrt(h*x)*a*g*x + 18*sqrt(h*x)*b*f*log(c) + (6*sqr
t(h*x)*h*x*log(e*x^2 + d) - (8*sqrt(h*x)*h*x/e - 6*sqrt(2)*(d*e^3*h^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^
2/e)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^4 - 6*sqrt(2)*(d*e^3*h^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h
^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/e^4 + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4
)*sqrt(h*x) + sqrt(d*h^2/e))/e^4 - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + s
qrt(d*h^2/e))/e^4)*e)*b*g*p/h + 18*sqrt(h*x)*a*f)/h

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{\sqrt {h\,x}} \,d x \]

[In]

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

[Out]

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