\(\int \frac {(-q+p x^2) \sqrt {q^2+p^2 x^4}}{x^2 (a q+b x+a p x^2)} \, dx\) [1916]

Optimal result

Integrand size = 43, antiderivative size = 133 \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\frac {\sqrt {q^2+p^2 x^4}}{a x}+\frac {2 \sqrt {-b^2+2 a^2 p q} \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x}{a q+b x+a p x^2+a \sqrt {q^2+p^2 x^4}}\right )}{a^2}+\frac {b \log (x)}{a^2}-\frac {b \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2} \]

[Out]

(p^2*x^4+q^2)^(1/2)/a/x+2*(2*a^2*p*q-b^2)^(1/2)*arctan((2*a^2*p*q-b^2)^(1/2)*x/(a*q+b*x+a*p*x^2+a*(p^2*x^4+q^2
)^(1/2)))/a^2+b*ln(x)/a^2-b*ln(q+p*x^2+(p^2*x^4+q^2)^(1/2))/a^2

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 3.76 (sec) , antiderivative size = 1209, normalized size of antiderivative = 9.09, number of steps used = 42, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {6860, 283, 311, 226, 1210, 272, 52, 65, 214, 1743, 1223, 1212, 1231, 1721, 1262, 749, 858, 223, 212, 739} \[ \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=-\frac {\text {arctanh}\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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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} \arctan \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 ) \text {arctanh}\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 ) \text {arctanh}\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}} \text {arctanh}\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}} \text {arctanh}\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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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} \]

[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 52

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 65

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 212

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

Rule 214

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

Rule 223

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 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 272

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 283

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 311

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 739

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 749

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 858

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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

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 1223

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 1231

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 1262

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 1721

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))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], 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 1743

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 6860

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*} \text {integral}& = \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 \text {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) \text {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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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) \text {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 \text {arctanh}\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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 \text {arctanh}\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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\frac {a \sqrt {q^2+p^2 x^4}+2 \sqrt {-b^2+2 a^2 p q} x \arctan \left (\frac {\sqrt {-b^2+2 a^2 p q} x}{b x+a \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}\right )+b x \log (x)-b x \log \left (q+p x^2+\sqrt {q^2+p^2 x^4}\right )}{a^2 x} \]

[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]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.10 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.53

method	result	size
pseudoelliptic	\(\frac {-b \ln \left (\frac {\sqrt {p^{2} x^{4}+q^{2}}+\left (p \,x^{2}+q \right ) \operatorname {csgn}\left (p \right )}{x}\right ) \operatorname {csgn}\left (p \right ) x a \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}+\sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, \sqrt {p^{2} x^{4}+q^{2}}\, a^{2}+2 \left (\ln \left (\frac {a \sqrt {p^{2} x^{4}+q^{2}}\, p \sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}+\left (-2 a q x -b \,x^{2}\right ) p^{2}-b p q}{\left (p \,x^{2}+q \right ) a +b x}\right )+\ln \left (2\right )\right ) x \left (a^{2} p q -\frac {b^{2}}{2}\right )}{\sqrt {\frac {-2 a^{2} p q +b^{2}}{a^{2}}}\, x \,a^{3}}\)	\(204\)
elliptic	\(\text {Expression too large to display}\)	\(1802\)
risch	\(\text {Expression too large to display}\)	\(6680\)
default	\(\text {Expression too large to display}\)	\(6985\)

[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]

2/((-2*a^2*p*q+b^2)/a^2)^(1/2)*(-1/2*b*ln(((p^2*x^4+q^2)^(1/2)+(p*x^2+q)*csgn(p))/x)*csgn(p)*x*a*((-2*a^2*p*q+
b^2)/a^2)^(1/2)+1/2*((-2*a^2*p*q+b^2)/a^2)^(1/2)*(p^2*x^4+q^2)^(1/2)*a^2+(ln((a*(p^2*x^4+q^2)^(1/2)*p*((-2*a^2
*p*q+b^2)/a^2)^(1/2)+(-2*a*q*x-b*x^2)*p^2-b*p*q)/((p*x^2+q)*a+b*x))+ln(2))*x*(a^2*p*q-1/2*b^2))/x/a^3

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\text {Timed out} \]

[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

Sympy [F]

\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int \frac {\left (p x^{2} - q\right ) \sqrt {p^{2} x^{4} + q^{2}}}{x^{2} \left (a p x^{2} + a q + b x\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int { \frac {\sqrt {p^{2} x^{4} + q^{2}} {\left (p x^{2} - q\right )}}{{\left (a p x^{2} + a q + b x\right )} x^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-q+p x^2\right ) \sqrt {q^2+p^2 x^4}}{x^2 \left (a q+b x+a p x^2\right )} \, dx=\int -\frac {\sqrt {p^2\,x^4+q^2}\,\left (q-p\,x^2\right )}{x^2\,\left (a\,p\,x^2+b\,x+a\,q\right )} \,d x \]

[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)