3.799 \(\int \frac {\sqrt {c+d x^4}}{x^2 (a+b x^4)} \, dx\)

Optimal. Leaf size=809 \[ \frac {b \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) c^{5/4}}{a (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \sqrt [4]{c}}{a \sqrt {d x^4+c}}-\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \tan ^{-1}\left (\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {d x^4+c}}\right )}{4 a}-\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \tan ^{-1}\left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {d x^4+c}}\right )}{4 a}-\frac {\sqrt {d x^4+c}}{a x}+\frac {\sqrt {d} x \sqrt {d x^4+c}}{a \left (\sqrt {d} x^2+\sqrt {c}\right )}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 \sqrt {b} \left (\sqrt {d} a^2+(-a)^{3/2} \sqrt {b} \sqrt {c}\right ) \sqrt [4]{d} \sqrt {d x^4+c} \sqrt [4]{c}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 a \sqrt {b} \left (\sqrt {d} a+\sqrt {-a} \sqrt {b} \sqrt {c}\right ) \sqrt [4]{d} \sqrt {d x^4+c} \sqrt [4]{c}} \]

[Out]

-(d*x^4+c)^(1/2)/a/x+x*d^(1/2)*(d*x^4+c)^(1/2)/a/(c^(1/2)+x^2*d^(1/2))-1/4*arctan(x*((a*d-b*c)/(-a)^(1/2)/b^(1
/2))^(1/2)/(d*x^4+c)^(1/2))*((a*d-b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/a-1/4*arctan(x*((-a*d+b*c)/(-a)^(1/2)/b^(1/2)
)^(1/2)/(d*x^4+c)^(1/2))*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/a-c^(1/4)*d^(1/4)*(cos(2*arctan(d^(1/4)*x/c^(1/
4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticE(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2
)+x^2*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/a/(d*x^4+c)^(1/2)+b*c^(5/4)*d^(1/4)*(cos(2*arctan(d^(
1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1
/2))*(c^(1/2)+x^2*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/a/(a*d+b*c)/(d*x^4+c)^(1/2)-1/8*(-a*d+b*c
)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*
x/c^(1/4))),1/4*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2
)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/c^(1/4)/d^(1/4)/
b^(1/2)/((-a)^(3/2)*b^(1/2)*c^(1/2)+a^2*d^(1/2))/(d*x^4+c)^(1/2)-1/8*(-a*d+b*c)*(cos(2*arctan(d^(1/4)*x/c^(1/4
)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),-1/4*c^(1/2)*(b^(1/2
)-(-a)^(1/2)*d^(1/2)/c^(1/2))^2/(-a)^(1/2)/b^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)
+(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/a/c^(1/4)/d^(1/4)/b^(1/2)/((-a)^(1/2)*b^(1/2)*c
^(1/2)+a*d^(1/2))/(d*x^4+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.51, antiderivative size = 1031, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {475, 584, 305, 220, 1196, 490, 1217, 1707} \[ -\frac {(b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 (-a)^{3/2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 (-a)^{5/4} \sqrt [4]{b}}+\frac {\sqrt {a d-b c} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 (-a)^{5/4} \sqrt [4]{b}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{a \sqrt {d x^4+c}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{2 a \sqrt {d x^4+c}}+\frac {\left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 (-a)^{3/2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {d x^4+c}}{a x}+\frac {\sqrt {d} x \sqrt {d x^4+c}}{a \left (\sqrt {d} x^2+\sqrt {c}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Int[Sqrt[c + d*x^4]/(x^2*(a + b*x^4)),x]

[Out]

-(Sqrt[c + d*x^4]/(a*x)) + (Sqrt[d]*x*Sqrt[c + d*x^4])/(a*(Sqrt[c] + Sqrt[d]*x^2)) + (Sqrt[b*c - a*d]*ArcTan[(
Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(5/4)*b^(1/4)) + (Sqrt[-(b*c) + a*d]*ArcTan[
(Sqrt[-(b*c) + a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(5/4)*b^(1/4)) - (c^(1/4)*d^(1/4)*(Sqrt[
c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(
a*Sqrt[c + d*x^4]) + (c^(1/4)*d^(1/4)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*Elli
pticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*a*Sqrt[c + d*x^4]) + ((Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(
1/4)*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/
4)*x)/c^(1/4)], 1/2])/(4*a*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1
/4)*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4
)*x)/c^(1/4)], 1/2])/(4*a*c^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) + ((Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(b*c
- a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqrt[c] - Sqrt
[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*(-a)^(3/2)*Sqrt[
b]*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4]) - ((Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(b*c - a*d)*(Sqrt[c]
+ Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/
(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*(-a)^(3/2)*Sqrt[b]*c^(1/4)*d^(1/
4)*(b*c + a*d)*Sqrt[c + d*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 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 475

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

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x^4}}{x^2 \left (a+b x^4\right )} \, dx &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\int \frac {x^2 \left (-b c+2 a d+b d x^4\right )}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{a}\\ &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\int \left (\frac {d x^2}{\sqrt {c+d x^4}}+\frac {(-b c+a d) x^2}{\left (a+b x^4\right ) \sqrt {c+d x^4}}\right ) \, dx}{a}\\ &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {d \int \frac {x^2}{\sqrt {c+d x^4}} \, dx}{a}+\frac {(-b c+a d) \int \frac {x^2}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{a}\\ &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\left (\sqrt {c} \sqrt {d}\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{a}-\frac {\left (\sqrt {c} \sqrt {d}\right ) \int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c}}}{\sqrt {c+d x^4}} \, dx}{a}+\frac {(b c-a d) \int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 a \sqrt {b}}-\frac {(b c-a d) \int \frac {1}{\left (\sqrt {-a}+\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 a \sqrt {b}}\\ &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\sqrt {d} x \sqrt {c+d x^4}}{a \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{a \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{2 a \sqrt {c+d x^4}}+\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 a (b c+a d)}-\frac {\left (\sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (\sqrt {-a}+\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 a (b c+a d)}+\frac {\left (\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 a \sqrt {b} (b c+a d)}+\frac {\left (\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 a \sqrt {b} (b c+a d)}\\ &=-\frac {\sqrt {c+d x^4}}{a x}+\frac {\sqrt {d} x \sqrt {c+d x^4}}{a \left (\sqrt {c}+\sqrt {d} x^2\right )}+\frac {\sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{5/4} \sqrt [4]{b}}+\frac {\sqrt {-b c+a d} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{5/4} \sqrt [4]{b}}-\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{a \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{2 a \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt {b} \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 a \sqrt {b} \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 (-a)^{3/2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 (-a)^{3/2} \sqrt {b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.13, size = 138, normalized size = 0.17 \[ \frac {-7 x^4 \sqrt {\frac {d x^4}{c}+1} (b c-2 a d) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+3 b d x^8 \sqrt {\frac {d x^4}{c}+1} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )-21 a \left (c+d x^4\right )}{21 a^2 x \sqrt {c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c + d*x^4]/(x^2*(a + b*x^4)),x]

[Out]

(-21*a*(c + d*x^4) - 7*(b*c - 2*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4
)/a)] + 3*b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(21*a^2*x*Sqrt[
c + d*x^4])

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fricas [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((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), x)

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maple [C]  time = 0.29, size = 421, normalized size = 0.52 \[ -\frac {\left (\frac {i \sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )\right ) \sqrt {c}\, \sqrt {d}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, b}-\frac {\left (a d -b c \right ) \left (\frac {2 \sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{3} b \EllipticPi \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , \frac {i \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{2} b \sqrt {c}}{a \sqrt {d}}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, a}-\frac {\arctanh \left (\frac {2 \RootOf \left (b \,\textit {\_Z}^{4}+a \right )^{2} d \,x^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}\right )}{8 b^{2} \RootOf \left (b \,\textit {\_Z}^{4}+a \right )}\right ) b}{a}+\frac {\frac {2 i \sqrt {-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \sqrt {\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, x , i\right )\right ) \sqrt {c}\, \sqrt {d}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\sqrt {d \,x^{4}+c}}{x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x)

[Out]

-1/a*b*(I*d^(1/2)/b*c^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2)*(-I/c^(1/2)*d^(1/2)*x^2+1)^(1/2)*(I/c^(1/2)*d^(1/2)*x^2+
1)^(1/2)/(d*x^4+c)^(1/2)*(EllipticF((I/c^(1/2)*d^(1/2))^(1/2)*x,I)-EllipticE((I/c^(1/2)*d^(1/2))^(1/2)*x,I))-1
/8/b^2*sum((a*d-b*c)/_alpha*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)/((-a*d+b*c)/b)^(1/2)/(
d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*(-I/c^(1/2)*d^(1/2)*x^2+1)^(1/2)*(I/c^(1/2)*d^(1/2)*x
^2+1)^(1/2)/(d*x^4+c)^(1/2)*EllipticPi((I/c^(1/2)*d^(1/2))^(1/2)*x,I*_alpha^2/a*b*c^(1/2)/d^(1/2),(-I/c^(1/2)*
d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a)))+1/a*(-(d*x^4+c)^(1/2)/x+2*I*d^(1/2)*c^(1/
2)/(I/c^(1/2)*d^(1/2))^(1/2)*(-I/c^(1/2)*d^(1/2)*x^2+1)^(1/2)*(I/c^(1/2)*d^(1/2)*x^2+1)^(1/2)/(d*x^4+c)^(1/2)*
(EllipticF((I/c^(1/2)*d^(1/2))^(1/2)*x,I)-EllipticE((I/c^(1/2)*d^(1/2))^(1/2)*x,I)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^(1/2)/x^2/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^4)^(1/2)/(x^2*(a + b*x^4)),x)

[Out]

int((c + d*x^4)^(1/2)/(x^2*(a + b*x^4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x^{4}}}{x^{2} \left (a + b x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**4+c)**(1/2)/x**2/(b*x**4+a),x)

[Out]

Integral(sqrt(c + d*x**4)/(x**2*(a + b*x**4)), x)

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