3.614 \(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{7/2}} \, dx\)

Optimal. Leaf size=935 \[ \frac {2 \sqrt {2} b e^{5/4} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{5 h (h x)^{5/2}}-\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {8 b e p f^2}{5 d h^3 \sqrt {h x}}-\frac {4 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{3 d^{3/4} h^{7/2}}-\frac {4 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{3 h^2 (h x)^{3/2}}-\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}} \]

[Out]

-2/5*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(5/2)-4/3*f*g*(a+b*ln(c*(e*x^2+d)^p))/h^2/(h*x)^(3/2)+2/5*b*e^(5/4)*f
^2*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(7/2)-4/3*b*e^(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)/d^(3/4)/h^(7/2)-2*b*e^(1/4)*g^2*p*arctan(1-e^(1/4)*2^(1
/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(1/4)/h^(7/2)-2/5*b*e^(5/4)*f^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1
/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(7/2)+4/3*b*e^(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)/d^(3/4)/h^(7/2)+2*b*e^(1/4)*g^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/
2)/d^(1/4)/h^(7/2)-1/5*b*e^(5/4)*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)/d^(5/4)/h^(7/2)-2/3*b*e^(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)/d^(3/4)/h^(7/2)+b*e^(1/4)*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)/d^(1/4)/h^(7/2)+1/5*b*e^(5/4)*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)/d^(5/4)/h^(7/2)+2/3*b*e^(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)/d^(3/4)/h^(7/2)-b*e^(1/4)*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)/d^(1/4)/h^(7/2)-8/5*b*e*f^2*p/d/h^3/(h*x)^(1/2)-2*g^2*(a+b*ln(c*(e*
x^2+d)^p))/h^3/(h*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.18, antiderivative size = 935, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2467, 2476, 2455, 325, 297, 1162, 617, 204, 1165, 628, 211} \[ \frac {2 \sqrt {2} b e^{5/4} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{5 h (h x)^{5/2}}-\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {8 b e p f^2}{5 d h^3 \sqrt {h x}}-\frac {4 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{3 d^{3/4} h^{7/2}}-\frac {4 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{3 h^2 (h x)^{3/2}}-\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]

[Out]

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

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 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 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]]))

Rule 325

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

Rule 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 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 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 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 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]

Rubi steps

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

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Mathematica [C]  time = 1.08, size = 340, normalized size = 0.36 \[ \frac {2 x^{7/2} \left (-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac {2 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}-\frac {\sqrt {2} b e^{3/4} f g p \left (\log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )-\log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )+2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}+1\right )\right )}{3 d^{3/4}}-\frac {4 b e f^2 p \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {e x^2}{d}\right )}{5 d \sqrt {x}}+\frac {2 b \sqrt [4]{e} g^2 p \left (\tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}\right )}{(h x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]

[Out]

(2*x^(7/2)*((2*b*e^(1/4)*g^2*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]
))/(-d)^(1/4) - (4*b*e*f^2*p*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)])/(5*d*Sqrt[x]) - (Sqrt[2]*b*e^(3/4)
*f*g*p*(2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + Lo
g[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] - Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqr
t[e]*x]))/(3*d^(3/4)) - (f^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*x^(5/2)) - (2*f*g*(a + b*Log[c*(d + e*x^2)^p]))/
(3*x^(3/2)) - (g^2*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(7/2)

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fricas [B]  time = 1.11, size = 2205, normalized size = 2.36 \[ \text {result too large to display} \]

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)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85
500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^
2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6
+ 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 + 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*
e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 1
0*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^2*h^7*sqrt(-(81*b^4*e
^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)
*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) - d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b
^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*
g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*
d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 - 32
*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 -
85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 10*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f
^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^
4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b
^2*d*e*f*g^3)*p^2)/(d^2*h^7))) - d*h^4*x^3*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 401
50*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g
- 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4
- 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 + 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(8
1*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4
*e*g^8)*p^4/(d^5*h^14)) + 10*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sq
rt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^
2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) + d*h^4*x
^3*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e
^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(3
2*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*
d^4*e*g^8)*sqrt(h*x)*p^3 - 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 +
40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 10*(9*b^2*d^2*
e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 342
0*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^
14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) + (10*a*d*f*g*x + 3*a*d*f^2 + 3*(4*b*e*f^2*p + 5*
a*d*g^2)*x^2 + (15*b*d*g^2*p*x^2 + 10*b*d*f*g*p*x + 3*b*d*f^2*p)*log(e*x^2 + d) + (15*b*d*g^2*x^2 + 10*b*d*f*g
*x + 3*b*d*f^2)*log(c))*sqrt(h*x))/(d*h^4*x^3)

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giac [A]  time = 0.56, size = 660, normalized size = 0.71 \[ \frac {\frac {2 \, {\left (10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + 15 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{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^{2} h} + \frac {2 \, {\left (10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + 15 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{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^{2} h} + \frac {{\left (10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - 15 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{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^{2} h} - \frac {{\left (10 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - 15 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{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^{2} h} - \frac {2 \, {\left (15 \, b d g^{2} h^{3} p x^{2} \log \left (h^{2} x^{2} e + d h^{2}\right ) - 15 \, b d g^{2} h^{3} p x^{2} \log \left (h^{2}\right ) + 12 \, b f^{2} h^{3} p x^{2} e + 10 \, b d f g h^{3} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 10 \, b d f g h^{3} p x \log \left (h^{2}\right ) + 15 \, b d g^{2} h^{3} x^{2} \log \relax (c) + 15 \, a d g^{2} h^{3} x^{2} + 3 \, b d f^{2} h^{3} p \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b d f^{2} h^{3} p \log \left (h^{2}\right ) + 10 \, b d f g h^{3} x \log \relax (c) + 10 \, a d f g h^{3} x + 3 \, b d f^{2} h^{3} \log \relax (c) + 3 \, a d f^{2} h^{3}\right )}}{\sqrt {h x} d h^{2} x^{2}}}{15 \, h^{4}} \]

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)^(7/2),x, algorithm="giac")

[Out]

1/15*(2*(10*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) + 15*sqrt(2)*(d*h^2)^(3/4)*b*d*g^2*p*e^(9/4) - 3*sqrt(2
)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*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^2*h) + 2*(10*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) + 15*sqrt(2)*(d*h^2)^(3/4)*b*d*
g^2*p*e^(9/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*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^2*h) + (10*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4) - 15*sqr
t(2)*(d*h^2)^(3/4)*b*d*g^2*p*e^(9/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*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^2*h) - (10*sqrt(2)*(d*h^2)^(1/4)*b*d*f*g*h*p*e^(11/4)
- 15*sqrt(2)*(d*h^2)^(3/4)*b*d*g^2*p*e^(9/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*f^2*p*e^(13/4))*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^2*h) - 2*(15*b*d*g^2*h^3*p*x^2*log(h^2*x^2*e
+ d*h^2) - 15*b*d*g^2*h^3*p*x^2*log(h^2) + 12*b*f^2*h^3*p*x^2*e + 10*b*d*f*g*h^3*p*x*log(h^2*x^2*e + d*h^2) -
10*b*d*f*g*h^3*p*x*log(h^2) + 15*b*d*g^2*h^3*x^2*log(c) + 15*a*d*g^2*h^3*x^2 + 3*b*d*f^2*h^3*p*log(h^2*x^2*e +
d*h^2) - 3*b*d*f^2*h^3*p*log(h^2) + 10*b*d*f*g*h^3*x*log(c) + 10*a*d*f*g*h^3*x + 3*b*d*f^2*h^3*log(c) + 3*a*d
*f^2*h^3)/(sqrt(h*x)*d*h^2*x^2))/h^4

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maple [F]  time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{2} \left (b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )+a \right )}{\left (h x \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(b*ln(c*(e*x^2+d)^p)+a)/(h*x)^(7/2),x)

[Out]

int((g*x+f)^2*(b*ln(c*(e*x^2+d)^p)+a)/(h*x)^(7/2),x)

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maxima [A]  time = 1.15, size = 813, normalized size = 0.87 \[ \text {result too large to display} \]

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)^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{7/2}} \,d x \]

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)^(7/2),x)

[Out]

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(7/2),x)

[Out]

Timed out

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