3.179 \(\int \frac {(a+b x^4)^{3/2}}{c+d x^4} \, dx\)

Optimal. Leaf size=926 \[ \frac {\sqrt [4]{b} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\sqrt [4]{b} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {b x^4+a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right ) (b c-a d)^{3/2}}{4 (-c)^{3/4} d^{7/4}}-\frac {(a d-b c)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{4 (-c)^{3/4} d^{7/4}}-\frac {b^{3/4} (3 b c-5 a d) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} d^2 \sqrt {b x^4+a}}+\frac {b x \sqrt {b x^4+a}}{3 d} \]

[Out]

-1/4*(-a*d+b*c)^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4)/d^(7/4)-1/4*(a*
d-b*c)^(3/2)*arctan(x*(a*d-b*c)^(1/2)/(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(3/4)/d^(7/4)+1/3*b*x*(b*x^4+a)
^(1/2)/d-1/6*b^(3/4)*(-5*a*d+3*b*c)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4))
)*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2
))^2)^(1/2)/a^(1/4)/d^2/(b*x^4+a)^(1/2)+1/4*b^(1/4)*(-a*d+b*c)^2*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/co
s(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*(
b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/d^2/(a*d+b*c)/(-c)^(1/2)
/(b*x^4+a)^(1/2)+1/8*(-a*d+b*c)^2*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*
EllipticPi(sin(2*arctan(b^(1/4)*x/a^(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))*(a^(1/2)+x^2*b^(1/2))*(b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))^2*((b*x^4+a)/(a^(1/2)+x^2*
b^(1/2))^2)^(1/2)/a^(1/4)/b^(1/4)/c/d^2/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4*b^(1/4)*(-a*d+b*c)^2*(cos(2*arctan(b^(1/
4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2
))*(a^(1/2)+x^2*b^(1/2))*(b^(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4
)/d^2/(a*d+b*c)/(-c)^(1/2)/(b*x^4+a)^(1/2)+1/8*(-a*d+b*c)^2*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*a
rctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(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))*(a^(1/2)+x^2*b^(1/2))*(b^(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))
^2*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/b^(1/4)/c/d^2/(a*d+b*c)/(b*x^4+a)^(1/2)

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Rubi [A]  time = 1.66, antiderivative size = 926, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {416, 523, 220, 409, 1217, 1707} \[ \frac {\sqrt [4]{b} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\sqrt [4]{b} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {b x^4+a}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) (b c-a d)^2}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {b x^4+a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right ) (b c-a d)^{3/2}}{4 (-c)^{3/4} d^{7/4}}-\frac {(a d-b c)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {b x^4+a}}\right )}{4 (-c)^{3/4} d^{7/4}}-\frac {b^{3/4} (3 b c-5 a d) \left (\sqrt {b} x^2+\sqrt {a}\right ) \sqrt {\frac {b x^4+a}{\left (\sqrt {b} x^2+\sqrt {a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} d^2 \sqrt {b x^4+a}}+\frac {b x \sqrt {b x^4+a}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/2)/(c + d*x^4),x]

[Out]

(b*x*Sqrt[a + b*x^4])/(3*d) - ((b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4
])])/(4*(-c)^(3/4)*d^(7/4)) - ((-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a +
b*x^4])])/(4*(-c)^(3/4)*d^(7/4)) - (b^(3/4)*(3*b*c - 5*a*d)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a]
+ Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(6*a^(1/4)*d^2*Sqrt[a + b*x^4]) + (b^(1/4)*(
Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*(b*c - a*d)^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*
x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*Sqrt[-c]*d^2*(b*c + a*d)*Sqrt[a + b*x^4]) +
(b^(1/4)*(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*(b*c - a*d)^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a]
+ Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*Sqrt[-c]*d^2*(b*c + a*d)*Sqrt[a +
b*x^4]) + ((Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])^2*(b*c - a*d)^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt
[a] + Sqrt[b]*x^2)^2]*EllipticPi[-(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]),
2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(8*a^(1/4)*b^(1/4)*c*d^2*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-
c] - Sqrt[a]*Sqrt[d])^2*(b*c - a*d)^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elli
pticPi[(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d]), 2*ArcTan[(b^(1/4)*x)/a^(1/
4)], 1/2])/(8*a^(1/4)*b^(1/4)*c*d^2*(b*c + a*d)*Sqrt[a + b*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 409

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

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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 {\left (a+b x^4\right )^{3/2}}{c+d x^4} \, dx &=\frac {b x \sqrt {a+b x^4}}{3 d}+\frac {\int \frac {-a (b c-3 a d)-b (3 b c-5 a d) x^4}{\sqrt {a+b x^4} \left (c+d x^4\right )} \, dx}{3 d}\\ &=\frac {b x \sqrt {a+b x^4}}{3 d}-\frac {(b (3 b c-5 a d)) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{3 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )} \, dx}{d^2}\\ &=\frac {b x \sqrt {a+b x^4}}{3 d}-\frac {b^{3/4} (3 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} d^2 \sqrt {a+b x^4}}+\frac {(b c-a d)^2 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c d^2}+\frac {(b c-a d)^2 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c d^2}\\ &=\frac {b x \sqrt {a+b x^4}}{3 d}-\frac {b^{3/4} (3 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} d^2 \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 \sqrt {-c} d^2 (b c+a d)}+\frac {\left (\sqrt {b} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{2 d^2 (b c+a d)}-\frac {\left (\sqrt {a} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (b c-a d)^2\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c d^{3/2} (b c+a d)}+\frac {\left (\sqrt {a} \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) (b c-a d)^2\right ) \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a}}}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {a+b x^4}} \, dx}{2 c d^{3/2} (b c+a d)}\\ &=\frac {b x \sqrt {a+b x^4}}{3 d}-\frac {(b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} d^{7/4}}-\frac {(-b c+a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 (-c)^{3/4} d^{7/4}}-\frac {b^{3/4} (3 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} d^2 \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (b c-a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt {-c} d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\sqrt [4]{b} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}\right ) (b c-a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2 (b c-a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} 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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2 (b c-a d)^2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} 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]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c d^2 (b c+a d) \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.52, size = 346, normalized size = 0.37 \[ \frac {x \left (\frac {5 \left (2 b x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c \left (3 a^2 d+a b d x^4+b^2 x^4 \left (c+d x^4\right )\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )-5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}+\frac {b x^4 \sqrt {\frac {b x^4}{a}+1} (5 a d-3 b c) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}\right )}{15 d \sqrt {a+b x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^4)^(3/2)/(c + d*x^4),x]

[Out]

(x*((b*(-3*b*c + 5*a*d)*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])/c + (5
*(-5*a*c*(3*a^2*d + a*b*d*x^4 + b^2*x^4*(c + d*x^4))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] +
2*b*x^4*(a + b*x^4)*(c + d*x^4)*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5
/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((c + d*x^4)*(-5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a),
-((d*x^4)/c)] + 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1
, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(15*d*Sqrt[a + b*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((b*x^4+a)^(3/2)/(d*x^4+c),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/2)/(d*x^4+c),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)/(d*x^4 + c), x)

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maple [C]  time = 0.48, size = 322, normalized size = 0.35 \[ \frac {\sqrt {b \,x^{4}+a}\, b x}{3 d}+\frac {\left (-\frac {a b}{3 d}+\frac {\left (2 a d -b c \right ) b}{d^{2}}\right ) \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \left (\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}+c \right )^{3} d \EllipticPi \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , \frac {i \RootOf \left (d \,\textit {\_Z}^{4}+c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {2 \RootOf \left (d \,\textit {\_Z}^{4}+c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{8 d^{3} \RootOf \left (d \,\textit {\_Z}^{4}+c \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/2)/(d*x^4+c),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/2)/(d*x^4+c),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)/(d*x^4 + c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}}{d\,x^4+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(3/2)/(c + d*x^4),x)

[Out]

int((a + b*x^4)^(3/2)/(c + d*x^4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{c + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/2)/(d*x**4+c),x)

[Out]

Integral((a + b*x**4)**(3/2)/(c + d*x**4), x)

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