3.20.16 \(\int \frac {(-q+p x^2) \sqrt {q^2+p^2 x^4}}{x^2 (a q+b x+a p x^2)} \, dx\)

Optimal. Leaf size=133 \[ \frac {2 \sqrt {2 a^2 p q-b^2} \tan ^{-1}\left (\frac {x \sqrt {2 a^2 p q-b^2}}{a \sqrt {p^2 x^4+q^2}+a p x^2+a q+b x}\right )}{a^2}-\frac {b \log \left (\sqrt {p^2 x^4+q^2}+p x^2+q\right )}{a^2}+\frac {b \log (x)}{a^2}+\frac {\sqrt {p^2 x^4+q^2}}{a x} \]

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Rubi [C]  time = 7.11, antiderivative size = 1209, normalized size of antiderivative = 9.09, number of steps used = 42, number of rules used = 20, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {6728, 277, 305, 220, 1196, 266, 50, 63, 208, 1729, 1209, 1198, 1217, 1707, 1248, 735, 844, 217, 206, 725} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {p^2 x^4+q^2}}{q}\right ) b}{2 a^2}+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) b}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right ) b}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2}}+\frac {\sqrt {p^2 x^4+q^2} b}{2 a^2 q}+\frac {\sqrt {2 a^2 p q-b^2} \tan ^{-1}\left (\frac {\sqrt {2 a^2 p q-b^2} x}{a \sqrt {p^2 x^4+q^2}}\right )}{a^2}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {p^2 x^4+q^2}}\right )}{4 a^2}+\frac {\sqrt {b^2-2 a^2 p q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2-b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}+\frac {\sqrt {b^2-2 a^2 p q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {-2 p q a^2+b^2+b \sqrt {b^2-4 a^2 p q}} \sqrt {p^2 x^4+q^2}}\right )}{4 \sqrt {2} a^4 p q}-\frac {\sqrt {p} \sqrt {q} \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {p^2 x^4+q^2}}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p^2 x^4+q^2}}{4 a^2 q}+\frac {\sqrt {p^2 x^4+q^2}}{a x}-\frac {\left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2} b}-\frac {\left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (p x^2+q\right ) \sqrt {\frac {p^2 x^4+q^2}{\left (p x^2+q\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {p^2 x^4+q^2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(x^2*(a*q + b*x + a*p*x^2)),x]

[Out]

(b*Sqrt[q^2 + p^2*x^4])/(2*a^2*q) - ((b - Sqrt[b^2 - 4*a^2*p*q])*Sqrt[q^2 + p^2*x^4])/(4*a^2*q) - ((b + Sqrt[b
^2 - 4*a^2*p*q])*Sqrt[q^2 + p^2*x^4])/(4*a^2*q) + Sqrt[q^2 + p^2*x^4]/(a*x) + (Sqrt[-b^2 + 2*a^2*p*q]*ArcTan[(
Sqrt[-b^2 + 2*a^2*p*q]*x)/(a*Sqrt[q^2 + p^2*x^4])])/a^2 - ((b - Sqrt[b^2 - 4*a^2*p*q])*ArcTanh[(p*x^2)/Sqrt[q^
2 + p^2*x^4]])/(4*a^2) - ((b + Sqrt[b^2 - 4*a^2*p*q])*ArcTanh[(p*x^2)/Sqrt[q^2 + p^2*x^4]])/(4*a^2) + (Sqrt[b^
2 - 2*a^2*p*q]*(b + Sqrt[b^2 - 4*a^2*p*q])*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p*q]]*ArcTanh[(p*(4*a^2*q
^2 + (b - Sqrt[b^2 - 4*a^2*p*q])^2*x^2))/(2*Sqrt[2]*Sqrt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 -
4*a^2*p*q]]*Sqrt[q^2 + p^2*x^4])])/(4*Sqrt[2]*a^4*p*q) + (Sqrt[b^2 - 2*a^2*p*q]*(b - Sqrt[b^2 - 4*a^2*p*q])*Sq
rt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]]*ArcTanh[(p*(4*a^2*q^2 + (b + Sqrt[b^2 - 4*a^2*p*q])^2*x^2))/(2*S
qrt[2]*Sqrt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]]*Sqrt[q^2 + p^2*x^4])])/(4*Sqrt[2]
*a^4*p*q) - (b*ArcTanh[Sqrt[q^2 + p^2*x^4]/q])/(2*a^2) - (Sqrt[p]*Sqrt[q]*(q + p*x^2)*Sqrt[(q^2 + p^2*x^4)/(q
+ p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(a*Sqrt[q^2 + p^2*x^4]) + (b*(b - Sqrt[b^2 - 4*a^2*
p*q])*(q + p*x^2)*Sqrt[(q^2 + p^2*x^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(4*a^3*Sq
rt[p]*Sqrt[q]*Sqrt[q^2 + p^2*x^4]) - ((b^2 - 2*a^2*p*q)*(b - Sqrt[b^2 - 4*a^2*p*q])*(q + p*x^2)*Sqrt[(q^2 + p^
2*x^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(4*a^3*b*Sqrt[p]*Sqrt[q]*Sqrt[q^2 + p^2*x
^4]) + (b*(b + Sqrt[b^2 - 4*a^2*p*q])*(q + p*x^2)*Sqrt[(q^2 + p^2*x^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt
[p]*x)/Sqrt[q]], 1/2])/(4*a^3*Sqrt[p]*Sqrt[q]*Sqrt[q^2 + p^2*x^4]) - ((b^2 - 2*a^2*p*q)*(b + Sqrt[b^2 - 4*a^2*
p*q])*(q + p*x^2)*Sqrt[(q^2 + p^2*x^4)/(q + p*x^2)^2]*EllipticF[2*ArcTan[(Sqrt[p]*x)/Sqrt[q]], 1/2])/(4*a^3*b*
Sqrt[p]*Sqrt[q]*Sqrt[q^2 + p^2*x^4])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rule 208

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 277

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

Rule 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1),
x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1198

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int
[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a
, c, d, e}, x] && PosQ[c/a]

Rule 1209

Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a +
c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c,
d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 0]

Rule 1217

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1707

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]),
x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1729

Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x
], x] - Dist[e, Int[(x*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx &=\int \left (-\frac {\sqrt {q^2+p^2 x^4}}{a x^2}+\frac {b \sqrt {q^2+p^2 x^4}}{a^2 q x}+\frac {\left (-b^2+2 a^2 p q-a b p x\right ) \sqrt {q^2+p^2 x^4}}{a^2 q \left (a q+b x+a p x^2\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {q^2+p^2 x^4}}{x^2} \, dx}{a}+\frac {\int \frac {\left (-b^2+2 a^2 p q-a b p x\right ) \sqrt {q^2+p^2 x^4}}{a q+b x+a p x^2} \, dx}{a^2 q}+\frac {b \int \frac {\sqrt {q^2+p^2 x^4}}{x} \, dx}{a^2 q}\\ &=\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {\left (2 p^2\right ) \int \frac {x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {\int \left (\frac {\left (-a b p-a p \sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x}+\frac {\left (-a b p+a p \sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x}\right ) \, dx}{a^2 q}+\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x}}{x} \, dx,x,x^4\right )}{4 a^2 q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {(b q) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {q^2+p^2 x}} \, dx,x,x^4\right )}{4 a^2}-\frac {(2 p q) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {(2 p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{a}-\frac {\left (p \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x} \, dx}{a q}-\frac {\left (p \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x} \, dx}{a q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-\left (4 a p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx-\left (4 a p^2\right ) \int \frac {\sqrt {q^2+p^2 x^4}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx+\frac {(b q) \operatorname {Subst}\left (\int \frac {1}{-\frac {q^2}{p^2}+\frac {x^2}{p^2}} \, dx,x,\sqrt {q^2+p^2 x^4}\right )}{2 a^2 p^2}+\frac {\left (2 p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx}{q}+\frac {\left (2 p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {x \sqrt {q^2+p^2 x^4}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2} \, dx}{q}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}+\frac {\int \frac {p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2+4 a^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 a^3 p^2}+\frac {\int \frac {p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2+4 a^2 p^4 x^2}{\sqrt {q^2+p^2 x^4}} \, dx}{4 a^3 p^2}+\frac {\left (p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x} \, dx,x,x^2\right )}{q}+\frac {\left (p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {q^2+p^2 x^2}}{\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x} \, dx,x,x^2\right )}{q}-\frac {1}{4} \left (a p^2 \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx-\frac {1}{4} \left (a p^2 \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-2 \frac {(p q) \int \frac {1-\frac {p x^2}{q}}{\sqrt {q^2+p^2 x^4}} \, dx}{a}+\frac {\left (b \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 a^3}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \operatorname {Subst}\left (\int \frac {-4 a^2 p^2 q^2-p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2 q}+\frac {\left (b \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{2 a^3}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \operatorname {Subst}\left (\int \frac {-4 a^2 p^2 q^2-p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2 q}-\frac {\left (a p^2 \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 b \left (b-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a^3 p^3 q \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 b \left (b-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a p^2 \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1}{\sqrt {q^2+p^2 x^4}} \, dx}{8 b \left (b+\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (a^3 p^3 q \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right )\right ) \int \frac {1+\frac {p x^2}{q}}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x^2\right ) \sqrt {q^2+p^2 x^4}} \, dx}{2 b \left (b+\sqrt {b^2-4 a^2 p q}\right )}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left (p \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (b-\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{a^2}-\frac {\left (p \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (b+\sqrt {b^2-4 a^2 p q}\right )^2-4 a^2 p^2 x\right ) \sqrt {q^2+p^2 x^2}} \, dx,x,x^2\right )}{a^2}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}-\frac {\left (p \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-p^2 x^2} \, dx,x,\frac {x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}-\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^4 p^4 q^2+p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 a^2 p^2 q^2-p^2 \left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {\left (p \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-p^2 x^2} \, dx,x,\frac {x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}-\frac {\left (p \left (b^2-2 a^2 p q\right ) \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 a^4 p^4 q^2+p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^4-x^2} \, dx,x,\frac {-4 a^2 p^2 q^2-p^2 \left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2}{\sqrt {q^2+p^2 x^4}}\right )}{a^2}\\ &=\frac {b \sqrt {q^2+p^2 x^4}}{2 a^2 q}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}{4 a^2 q}+\frac {\sqrt {q^2+p^2 x^4}}{a x}-\frac {2 p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a \sqrt {q^2+p^2 x^4}}\right )}{a^2}-\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}-\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p x^2}{\sqrt {q^2+p^2 x^4}}\right )}{4 a^2}+\frac {\sqrt {b^2-2 a^2 p q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b-\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {b^2-2 a^2 p q-b \sqrt {b^2-4 a^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{2 \sqrt {2} a^2 \sqrt {b^2-2 a^2 p q-b \sqrt {b^2-4 a^2 p q}}}+\frac {\sqrt {b^2-2 a^2 p q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \tanh ^{-1}\left (\frac {p \left (4 a^2 q^2+\left (b+\sqrt {b^2-4 a^2 p q}\right )^2 x^2\right )}{2 \sqrt {2} \sqrt {b^2-2 a^2 p q} \sqrt {b^2-2 a^2 p q+b \sqrt {b^2-4 a^2 p q}} \sqrt {q^2+p^2 x^4}}\right )}{2 \sqrt {2} a^2 \sqrt {b^2-2 a^2 p q+b \sqrt {b^2-4 a^2 p q}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {q^2+p^2 x^4}}{q}\right )}{2 a^2}+\frac {2 \sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}-2 \left (-\frac {p x \sqrt {q^2+p^2 x^4}}{a \left (q+p x^2\right )}+\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}\right )-\frac {\sqrt {p} \sqrt {q} \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{a \sqrt {q^2+p^2 x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{4 a^3 \sqrt {p} \sqrt {q} \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}-\frac {a p^{3/2} \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q-b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b-\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}+\frac {a p^{3/2} \left (b^2-4 a^2 p q+b \sqrt {b^2-4 a^2 p q}\right ) \left (16 q^2+\frac {\left (b+\sqrt {b^2-4 a^2 p q}\right )^4}{a^4 p^2}\right ) \left (q+p x^2\right ) \sqrt {\frac {q^2+p^2 x^4}{\left (q+p x^2\right )^2}} \Pi \left (\frac {b^2}{4 a^2 p q};2 \tan ^{-1}\left (\frac {\sqrt {p} x}{\sqrt {q}}\right )|\frac {1}{2}\right )}{16 b \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )^3 \sqrt {q^2+p^2 x^4}}\\ \end {align*}

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Mathematica [C]  time = 8.74, size = 3835, normalized size = 28.83 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(x^2*(a*q + b*x + a*p*x^2)),x]

[Out]

Sqrt[q^2 + p^2*x^4]/(a*x) - ((b*(2*ArcTanh[(p*x^2)/Sqrt[q^2 + p^2*x^4]] + 2*ArcTanh[Sqrt[q^2 + p^2*x^4]/q] + (
Sqrt[2]*Sqrt[b^2 - 2*a^2*p*q]*(b^2 - 4*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p*q])*ArcTanh[(p*(b*(b - Sqrt[b^2 - 4*a^2*
p*q])*x^2 + 2*a^2*q*(q - p*x^2)))/(Sqrt[2]*Sqrt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p*q
]]*Sqrt[q^2 + p^2*x^4])])/(b*Sqrt[b^2 - 4*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q - b*Sqrt[b^2 - 4*a^2*p*q]]) - (Sqrt[2]
*Sqrt[b^2 - 2*a^2*p*q]*(b^2 - 4*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q])*ArcTanh[(p*(b*(b + Sqrt[b^2 - 4*a^2*p*q])*x
^2 + 2*a^2*q*(q - p*x^2)))/(Sqrt[2]*Sqrt[b^2 - 2*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]]*Sqrt
[q^2 + p^2*x^4])])/(b*Sqrt[b^2 - 4*a^2*p*q]*Sqrt[b^2 - 2*a^2*p*q + b*Sqrt[b^2 - 4*a^2*p*q]])))/(4*a) + (I*b^2*
Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticF[I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^2*Sqrt[(I*p)/q]*Sqrt
[q^2 + p^2*x^4]) - ((2*I)*p*q*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticF[I*ArcSinh[Sqrt[(I*p)/q]*x]
, -1])/(Sqrt[(I*p)/q]*Sqrt[q^2 + p^2*x^4]) + (I*b^6*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-
2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^4*p^2*Sqrt[(
I*p)/q]*(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])*(-1/2*(b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(a^
2*p^2) + (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) - ((6*I)*b^4*q*Sqrt[1
- (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q
]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^2*p*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])*(-1/2
*(b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(a^2*p^2) + (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^
2*p^2))*Sqrt[q^2 + p^2*x^4]) + ((8*I)*b^2*q^2*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a
^2*p*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(Sqrt[(I*p)/q]*(-b^2
+ 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])*(-1/2*(b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(a^2*p^2) + (b^2
- 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) + (I*b^4*Sqrt[b^4 - 4*a^2*b^2*p*q]*
Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b
^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^4*p^2*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q
])*(-1/2*(b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(a^2*p^2) + (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q
])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) - ((4*I)*b^2*q*Sqrt[b^4 - 4*a^2*b^2*p*q]*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (
I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q
]*x], -1])/(a^2*p*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])*(-1/2*(b^2 - 2*a^2*p*q - Sqrt[b
^4 - 4*a^2*b^2*p*q])/(a^2*p^2) + (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4
]) + ((4*I)*q^2*Sqrt[b^4 - 4*a^2*b^2*p*q]*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p
*q)/(-b^2 + 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(Sqrt[(I*p)/q]*(-b^2 + 2*
a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])*(-1/2*(b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(a^2*p^2) + (b^2 - 2*
a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) + (I*b^6*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 +
(I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)
/q]*x], -1])/(a^4*p^2*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p*q - Sqrt[b^
4 - 4*a^2*b^2*p*q])/(2*a^2*p^2) - (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^
4]) - ((6*I)*b^4*q*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q +
Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^2*p*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q + Sqrt[b^
4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2) - (b^2 - 2*a^2*p*q + Sqrt[b^4 -
4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) + ((8*I)*b^2*q^2*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q
]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])
/(Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/
(2*a^2*p^2) - (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) - (I*b^4*Sqrt[b^
4 - 4*a^2*b^2*p*q]*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q +
Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(a^4*p^2*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q + Sqrt[
b^4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2) - (b^2 - 2*a^2*p*q + Sqrt[b^4
- 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]) + ((4*I)*b^2*q*Sqrt[b^4 - 4*a^2*b^2*p*q]*Sqrt[1 - (I*p*x^
2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcS
inh[Sqrt[(I*p)/q]*x], -1])/(a^2*p*Sqrt[(I*p)/q]*(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p
*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2) - (b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[
q^2 + p^2*x^4]) - ((4*I)*q^2*Sqrt[b^4 - 4*a^2*b^2*p*q]*Sqrt[1 - (I*p*x^2)/q]*Sqrt[1 + (I*p*x^2)/q]*EllipticPi[
((-2*I)*a^2*p*q)/(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q]), I*ArcSinh[Sqrt[(I*p)/q]*x], -1])/(Sqrt[(I*p)/
q]*(-b^2 + 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])*((b^2 - 2*a^2*p*q - Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2) -
(b^2 - 2*a^2*p*q + Sqrt[b^4 - 4*a^2*b^2*p*q])/(2*a^2*p^2))*Sqrt[q^2 + p^2*x^4]))/a

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IntegrateAlgebraic [A]  time = 1.56, size = 133, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {2 \sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt {q^2+p^2 x^4}}\right )}{a^2}+\frac {b \log (x)}{a^2}-\frac {b \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-q + p*x^2)*Sqrt[q^2 + p^2*x^4])/(x^2*(a*q + b*x + a*p*x^2)),x]

[Out]

Sqrt[q^2 + p^2*x^4]/(a*x) + (2*Sqrt[-b^2 + 2*a^2*p*q]*ArcTan[(Sqrt[-b^2 + 2*a^2*p*q]*x)/(a*q + b*x + a*p*x^2 +
a*Sqrt[q^2 + p^2*x^4])])/a^2 + (b*Log[x])/a^2 - (b*Log[q + p*x^2 + Sqrt[q^2 + p^2*x^4]])/a^2

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/x^2/(a*p*x^2+a*q+b*x),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/x^2/(a*p*x^2+a*q+b*x),x, algorithm="giac")

[Out]

integrate(sqrt(p^2*x^4 + q^2)*(p*x^2 - q)/((a*p*x^2 + a*q + b*x)*x^2), x)

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maple [C]  time = 0.67, size = 6680, normalized size = 50.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(6680\)
default	\(\text {Expression too large to display}\)	\(6985\)
elliptic	\(\text {Expression too large to display}\)	\(9277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/x^2/(a*p*x^2+a*q+b*x),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^2-q)*(p^2*x^4+q^2)^(1/2)/x^2/(a*p*x^2+a*q+b*x),x, algorithm="maxima")

[Out]

integrate(sqrt(p^2*x^4 + q^2)*(p*x^2 - q)/((a*p*x^2 + a*q + b*x)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )}{x^2\,\left (a\,p\,x^2+b\,x+a\,q\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((p^2*x^4 + q^2)^(1/2)*(q - p*x^2))/(x^2*(a*q + b*x + a*p*x^2)),x)

[Out]

int(-((p^2*x^4 + q^2)^(1/2)*(q - p*x^2))/(x^2*(a*q + b*x + a*p*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p x^{2} - q\right ) \sqrt {p^{2} x^{4} + q^{2}}}{x^{2} \left (a p x^{2} + a q + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x**2-q)*(p**2*x**4+q**2)**(1/2)/x**2/(a*p*x**2+a*q+b*x),x)

[Out]

Integral((p*x**2 - q)*sqrt(p**2*x**4 + q**2)/(x**2*(a*p*x**2 + a*q + b*x)), x)

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