3.7.11 \(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{\sqrt {h x}} \, dx\) [611]

Optimal. Leaf size=1002 \[ \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{5/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}} \]

[Out]

-16/9*b*f*g*p*(h*x)^(3/2)/h^2-8/25*b*g^2*p*(h*x)^(5/2)/h^3+4/3*f*g*(h*x)^(3/2)*(a+b*ln(c*(e*x^2+d)^p))/h^2+2/5
*g^2*(h*x)^(5/2)*(a+b*ln(c*(e*x^2+d)^p))/h^3-2*b*d^(1/4)*f^2*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)-4/3*b*d^(3/4)*f*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)+2/5*b*d^(5/4)*g^2*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(5/4)/h
^(1/2)+2*b*d^(1/4)*f^2*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)+4/3*b*d
^(3/4)*f*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)-2/5*b*d^(5/4)*g^2*p
*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/e^(5/4)/h^(1/2)-b*d^(1/4)*f^2*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)+2/3*b*d^(3/4)*f*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)+1/5*b*d^(5/4)*g^2*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^(5/4)/h^(1/2)+b*d^(1/4)*f
^2*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)-2/3*b*d
^(3/4)*f*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)
-1/5*b*d^(5/4)*g^2*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^(5/4)
/h^(1/2)+2*a*f^2*(h*x)^(1/2)/h-8*b*f^2*p*(h*x)^(1/2)/h+8/5*b*d*g^2*p*(h*x)^(1/2)/e/h+2*b*f^2*ln(c*(e*x^2+d)^p)
*(h*x)^(1/2)/h

________________________________________________________________________________________

Rubi [A]
time = 0.88, antiderivative size = 1002, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2517, 2521, 2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303, 308} \begin {gather*} -\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{5/2}}{5 h^3}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}+\frac {4 f g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{3/2}}{3 h^2}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}+\frac {2 b f^2 \log \left (c \left (e x^2+d\right )^p\right ) \sqrt {h x}}{h}+\frac {2 a f^2 \sqrt {h x}}{h}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \text {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^{5/4} g^2 p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \text {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^2 p \text {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 \sqrt {2} b d^{5/4} g^2 p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{5 e^{5/4} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \text {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 {\sqrt {2} b \sqrt [4]{d} f^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 308

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

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

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Mathematica [A]
time = 0.90, size = 588, normalized size = 0.59 \begin {gather*} \frac {2 \sqrt {x} \left (a f^2 \sqrt {x}-\frac {4 b f g p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}-\frac {b f^2 p \left (8 \sqrt [4]{e} \sqrt {x}+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 (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{2 \sqrt [4]{e}}-\frac {b g^2 p \left (-40 d \sqrt [4]{e} \sqrt {x}+8 e^{5/4} x^{5/2}-10 \sqrt {2} d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+10 \sqrt {2} d^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )+5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{50 e^{5/4}}+b f^2 \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+\frac {1}{5} g^2 x^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{\sqrt {h x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(a*f^2*Sqrt[x] - (4*b*f*g*p*(2*(-d)^(1/4)*e^(3/4)*x^(3/2) - 3*d*ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)
] + 3*d*ArcTanh[(e^(1/4)*Sqrt[x])/(-d)^(1/4)]))/(9*(-d)^(1/4)*e^(3/4)) - (b*f^2*p*(8*e^(1/4)*Sqrt[x] + 2*Sqrt[
2]*d^(1/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 2*Sqrt[2]*d^(1/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[
x])/d^(1/4)] + Sqrt[2]*d^(1/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] - Sqrt[2]*d^(1/4)*Lo
g[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(2*e^(1/4)) - (b*g^2*p*(-40*d*e^(1/4)*Sqrt[x] + 8*e
^(5/4)*x^(5/2) - 10*Sqrt[2]*d^(5/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + 10*Sqrt[2]*d^(5/4)*ArcTan[
1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 5*Sqrt[2]*d^(5/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqr
t[e]*x] + 5*Sqrt[2]*d^(5/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x]))/(50*e^(5/4)) + b*f^2*
Sqrt[x]*Log[c*(d + e*x^2)^p] + (2*f*g*x^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/3 + (g^2*x^(5/2)*(a + b*Log[c*(d +
e*x^2)^p]))/5))/Sqrt[h*x]

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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\sqrt {h x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 847, normalized size = 0.85 \begin {gather*} \frac {2 \, b g^{2} x^{3} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{5 \, \sqrt {h x}} + \frac {2 \, a g^{2} x^{3}}{5 \, \sqrt {h x}} + \frac {4 \, b f g x^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, \sqrt {h x}} + \frac {4 \, a f g x^{2}}{3 \, \sqrt {h x}} + \frac {2 \, \sqrt {h x} b f^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{h} - \frac {{\left (8 \, \sqrt {h x} h^{2} e^{\left (-1\right )} - {\left (\frac {\sqrt {2} h^{4} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \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}}} - \frac {\sqrt {2} h^{4} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \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}}} + \frac {2 \, \sqrt {2} h^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}} + \frac {2 \, \sqrt {2} h^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}}\right )} d e^{\left (-1\right )}\right )} b f^{2} p e}{h^{3}} + \frac {2 \, \sqrt {h x} a f^{2}}{h} - \frac {2 \, {\left (3 \, {\left (\frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \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}}} - \frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \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}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}} - \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}}\right )} d h^{4} e^{\left (-1\right )} + 8 \, \left (h x\right )^{\frac {3}{2}} h^{2} e^{\left (-1\right )}\right )} b f g p e}{9 \, h^{4}} - \frac {{\left (5 \, {\left (\frac {\sqrt {2} h^{6} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \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}}} - \frac {\sqrt {2} h^{6} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \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}}} + \frac {2 \, \sqrt {2} h^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}} + \frac {2 \, \sqrt {2} h^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}}\right )} d^{2} e^{\left (-2\right )} - 8 \, {\left (5 \, \sqrt {h x} d h^{4} - \left (h x\right )^{\frac {5}{2}} h^{2} e\right )} e^{\left (-2\right )}\right )} b g^{2} p e}{25 \, h^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*g^2*x^3*log((x^2*e + d)^p*c)/sqrt(h*x) + 2/5*a*g^2*x^3/sqrt(h*x) + 4/3*b*f*g*x^2*log((x^2*e + d)^p*c)/sq
rt(h*x) + 4/3*a*f*g*x^2/sqrt(h*x) + 2*sqrt(h*x)*b*f^2*log((x^2*e + d)^p*c)/h - (8*sqrt(h*x)*h^2*e^(-1) - (sqrt
(2)*h^4*e^(-1/4)*log(h*x*e^(1/2) + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2)^(3/4) - sqrt(2
)*h^4*e^(-1/4)*log(h*x*e^(1/2) - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2)^(3/4) + 2*sqrt(2
)*h^3*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*e^(1/2))*e^(-1/4)/sqrt(sqrt(d)*h))*e^(-1
/4)/(sqrt(sqrt(d)*h)*sqrt(d)) + 2*sqrt(2)*h^3*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)
*e^(1/2))*e^(-1/4)/sqrt(sqrt(d)*h))*e^(-1/4)/(sqrt(sqrt(d)*h)*sqrt(d)))*d*e^(-1))*b*f^2*p*e/h^3 + 2*sqrt(h*x)*
a*f^2/h - 2/9*(3*(sqrt(2)*e^(-3/4)*log(h*x*e^(1/2) + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h
^2)^(1/4) - sqrt(2)*e^(-3/4)*log(h*x*e^(1/2) - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2)^(1
/4) - 2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*e^(1/2))*e^(-1/4)/sqrt(sqrt(d)
*h))*e^(-3/4)/sqrt(sqrt(d)*h) - 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*e^(
1/2))*e^(-1/4)/sqrt(sqrt(d)*h))*e^(-3/4)/sqrt(sqrt(d)*h))*d*h^4*e^(-1) + 8*(h*x)^(3/2)*h^2*e^(-1))*b*f*g*p*e/h
^4 - 1/25*(5*(sqrt(2)*h^6*e^(-1/4)*log(h*x*e^(1/2) + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h
^2)^(3/4) - sqrt(2)*h^6*e^(-1/4)*log(h*x*e^(1/2) - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/(d*h^2
)^(3/4) + 2*sqrt(2)*h^5*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*e^(1/2))*e^(-1/4)/sqrt
(sqrt(d)*h))*e^(-1/4)/(sqrt(sqrt(d)*h)*sqrt(d)) + 2*sqrt(2)*h^5*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(
1/4) - 2*sqrt(h*x)*e^(1/2))*e^(-1/4)/sqrt(sqrt(d)*h))*e^(-1/4)/(sqrt(sqrt(d)*h)*sqrt(d)))*d^2*e^(-2) - 8*(5*sq
rt(h*x)*d*h^4 - (h*x)^(5/2)*h^2*e)*e^(-2))*b*g^2*p*e/h^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2326 vs. \(2 (680) = 1360\).
time = 21.96, size = 2326, normalized size = 2.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/225*(15*h*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e + h*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f
^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)
/h^2)*e^2)/h)*log(16*(27*b^2*d^3*g^6*h*p^2*e - 705*b^2*d^2*f^2*g^4*h*p^2*e^2 + 3525*b^2*d*f^4*g^2*h*p^2*e^3 -
3375*b^2*f^6*h*p^2*e^4 + 10*f*g*h^2*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4
*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^4)*sqrt((60*b^2*d^2*f*g^
3*p^2 - 300*b^2*d*f^3*g*p^2*e + h*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g
^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/h)*e^(-1) + 16*(81*b^3*
d^4*g^8*p^3 - 1620*b^3*d^3*f^2*g^6*p^3*e + 2150*b^3*d^2*f^4*g^4*p^3*e^2 - 40500*b^3*d*f^6*g^2*p^3*e^3 + 50625*
b^3*f^8*p^3*e^4)*sqrt(h*x)) - 15*h*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e + h*sqrt(-(81*b^4*d^5*g^
8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4
*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/h)*log(-16*(27*b^2*d^3*g^6*h*p^2*e - 705*b^2*d^2*f^2*g^4*h*p^2*e^2 + 3525*b^2
*d*f^4*g^2*h*p^2*e^3 - 3375*b^2*f^6*h*p^2*e^4 + 10*f*g*h^2*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^
4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^4
)*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e + h*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*
e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/
h)*e^(-1) + 16*(81*b^3*d^4*g^8*p^3 - 1620*b^3*d^3*f^2*g^6*p^3*e + 2150*b^3*d^2*f^4*g^4*p^3*e^2 - 40500*b^3*d*f
^6*g^2*p^3*e^3 + 50625*b^3*f^8*p^3*e^4)*sqrt(h*x)) + 15*h*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e -
h*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*
g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/h)*log(16*(27*b^2*d^3*g^6*h*p^2*e - 705*b^2*d^2*f^2*g^
4*h*p^2*e^2 + 3525*b^2*d*f^4*g^2*h*p^2*e^3 - 3375*b^2*f^6*h*p^2*e^4 - 10*f*g*h^2*sqrt(-(81*b^4*d^5*g^8*p^4 - 3
420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^
4*e^4)*e^(-5)/h^2)*e^4)*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e - h*sqrt(-(81*b^4*d^5*g^8*p^4 - 342
0*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*
e^4)*e^(-5)/h^2)*e^2)/h)*e^(-1) + 16*(81*b^3*d^4*g^8*p^3 - 1620*b^3*d^3*f^2*g^6*p^3*e + 2150*b^3*d^2*f^4*g^4*p
^3*e^2 - 40500*b^3*d*f^6*g^2*p^3*e^3 + 50625*b^3*f^8*p^3*e^4)*sqrt(h*x)) - 15*h*sqrt((60*b^2*d^2*f*g^3*p^2 - 3
00*b^2*d*f^3*g*p^2*e - h*sqrt(-(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^
2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/h)*log(-16*(27*b^2*d^3*g^6*h*p^2
*e - 705*b^2*d^2*f^2*g^4*h*p^2*e^2 + 3525*b^2*d*f^4*g^2*h*p^2*e^3 - 3375*b^2*f^6*h*p^2*e^4 - 10*f*g*h^2*sqrt(-
(81*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e
^3 + 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^4)*sqrt((60*b^2*d^2*f*g^3*p^2 - 300*b^2*d*f^3*g*p^2*e - h*sqrt(-(8
1*b^4*d^5*g^8*p^4 - 3420*b^4*d^4*f^2*g^6*p^4*e + 40150*b^4*d^3*f^4*g^4*p^4*e^2 - 85500*b^4*d^2*f^6*g^2*p^4*e^3
+ 50625*b^4*d*f^8*p^4*e^4)*e^(-5)/h^2)*e^2)/h)*e^(-1) + 16*(81*b^3*d^4*g^8*p^3 - 1620*b^3*d^3*f^2*g^6*p^3*e +
2150*b^3*d^2*f^4*g^4*p^3*e^2 - 40500*b^3*d*f^6*g^2*p^3*e^3 + 50625*b^3*f^8*p^3*e^4)*sqrt(h*x)) - (180*b*d*g^2
*p + 15*(3*b*g^2*p*x^2 + 10*b*f*g*p*x + 15*b*f^2*p)*e*log(x^2*e + d) + 15*(3*b*g^2*x^2 + 10*b*f*g*x + 15*b*f^2
)*e*log(c) - (900*b*f^2*p - 225*a*f^2 + 9*(4*b*g^2*p - 5*a*g^2)*x^2 + 50*(4*b*f*g*p - 3*a*f*g)*x)*e)*sqrt(h*x)
)*e^(-1)/h

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.02, size = 820, normalized size = 0.82 \begin {gather*} \frac {90 \, \sqrt {h x} b g^{2} x^{2} \log \left (c\right ) + 90 \, \sqrt {h x} a g^{2} x^{2} + 300 \, \sqrt {h x} b f g x \log \left (c\right ) + 225 \, {\left ({\left (2 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \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 (-\frac {5}{4}\right )} + 2 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \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 (-\frac {5}{4}\right )} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {5}{4}\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 ) - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {5}{4}\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 ) - 8 \, \sqrt {h x} e^{\left (-1\right )}\right )} e + 2 \, \sqrt {h x} \log \left (x^{2} e + d\right )\right )} b f^{2} p + 9 \, {\left (10 \, \sqrt {h x} x^{2} \log \left (x^{2} e + d\right ) - {\left (10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} d \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 (-\frac {9}{4}\right )} + 10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} d \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 (-\frac {9}{4}\right )} + 5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} d e^{\left (-\frac {9}{4}\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 ) - 5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} d e^{\left (-\frac {9}{4}\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 ) + \frac {8 \, {\left (\sqrt {h x} h^{10} x^{2} e^{4} - 5 \, \sqrt {h x} d h^{10} e^{3}\right )} e^{\left (-5\right )}}{h^{10}}\right )} e\right )} b g^{2} p + 300 \, \sqrt {h x} a f g x + 450 \, \sqrt {h x} b f^{2} \log \left (c\right ) + \frac {50 \, {\left (6 \, \sqrt {h x} h x \log \left (x^{2} e + d\right ) - {\left (8 \, \sqrt {h x} h x e^{\left (-1\right )} - 6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} \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 (-\frac {7}{4}\right )} - 6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} \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 (-\frac {7}{4}\right )} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} e^{\left (-\frac {7}{4}\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 ) - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} e^{\left (-\frac {7}{4}\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 )\right )} e\right )} b f g p}{h} + 450 \, \sqrt {h x} a f^{2}}{225 \, h} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{\sqrt {h\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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