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

Optimal. Leaf size=524 \[ \frac {\left (\sqrt {2} b \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b}-i \sqrt {2} \sqrt {b} \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b}}{\sqrt {a} \left (\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}}-\frac {i \left (\sqrt {2} \sqrt {b} \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}-i \sqrt {2} b \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b}}{\sqrt {a} \left (\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}}+\frac {\sqrt {p x^4+q}}{a x} \]

________________________________________________________________________________________

Rubi [C]  time = 3.47, antiderivative size = 1029, normalized size of antiderivative = 1.96, number of steps used = 22, number of rules used = 9, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {6728, 277, 305, 220, 1196, 1209, 1198, 1217, 1707} \begin {gather*} \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {p x^4+q}}\right )}{a^{3/2}}-\frac {\left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q}}-\frac {\left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q}}-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {p x^4+q}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}-\frac {b \left (2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}-\frac {b \left (2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}+\frac {\sqrt {p x^4+q}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [C]  time = 1.51, size = 293, normalized size = 0.56 \begin {gather*} \frac {-i b x \sqrt {\frac {p x^4}{q}+1} \Pi \left (\frac {2 i a \sqrt {p} \sqrt {q}}{\sqrt {b^2-4 a^2 p q}-b};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )-i b x \sqrt {\frac {p x^4}{q}+1} \Pi \left (-\frac {2 i a \sqrt {p} \sqrt {q}}{b+\sqrt {b^2-4 a^2 p q}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )+a p x^4 \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}+a q \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}+i b x \sqrt {\frac {p x^4}{q}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )}{a^2 x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} \sqrt {p x^4+q}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[(I*Sqrt[p])/Sqrt[q]]*q + a*p*Sqrt[(I*Sqrt[p])/Sqrt[q]]*x^4 + I*b*x*Sqrt[1 + (p*x^4)/q]*EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[p])/Sqrt[q]]*x], -1] - I*b*x*Sqrt[1 + (p*x^4)/q]*EllipticPi[((2*I)*a*Sqrt[p]*Sqrt[q])/(-b +
Sqrt[b^2 - 4*a^2*p*q]), I*ArcSinh[Sqrt[(I*Sqrt[p])/Sqrt[q]]*x], -1] - I*b*x*Sqrt[1 + (p*x^4)/q]*EllipticPi[((-
2*I)*a*Sqrt[p]*Sqrt[q])/(b + Sqrt[b^2 - 4*a^2*p*q]), I*ArcSinh[Sqrt[(I*Sqrt[p])/Sqrt[q]]*x], -1])/(a^2*Sqrt[(I
*Sqrt[p])/Sqrt[q]]*x*Sqrt[q + p*x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.88, size = 54, normalized size = 0.10 \begin {gather*} \frac {\sqrt {q+p x^4}}{a x}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {q+p x^4}}\right )}{a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[q + p*x^4]/(a*x) + (Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^4])])/a^(3/2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.18, size = 60, normalized size = 0.11

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{a x}-\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {p \,x^{4}+q}\, a}{x \sqrt {a b}}\right )}{a \sqrt {a b}}\right ) \sqrt {2}}{2}\)	\(60\)
risch	\(\frac {\sqrt {p \,x^{4}+q}}{a x}-\frac {b \left (\frac {\sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )}{a \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} b -2 a q \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a p +a p \,x^{2}-b \right )}{a \sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}\, \sqrt {p \,x^{4}+q}}\right )}{\sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right ) \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}{\sqrt {p}\, \sqrt {q}\, a}, \frac {\sqrt {-\frac {i \sqrt {p}}{\sqrt {q}}}}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}}\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, a q \sqrt {p \,x^{4}+q}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}}{4 a}\right )}{a}\)	\(332\)
default	\(-\frac {-\frac {\sqrt {p \,x^{4}+q}}{x}+\frac {2 i \sqrt {p}\, \sqrt {q}\, \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}}{a}+\frac {-\frac {b \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )}{a \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}+\frac {2 i \sqrt {p}\, \sqrt {q}\, \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} b +2 a q \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a p +a p \,x^{2}-b \right )}{a \sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}\, \sqrt {p \,x^{4}+q}}\right )}{\sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right ) \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}{\sqrt {p}\, \sqrt {q}\, a}, \frac {\sqrt {-\frac {i \sqrt {p}}{\sqrt {q}}}}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}}\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, a q \sqrt {p \,x^{4}+q}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}\right )}{4 a}}{a}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________