3.183 \(\int \frac{1}{(a+b x^4)^{5/2} (c+d x^4)} \, dx\)

Optimal. Leaf size=976 \[ \text{result too large to display} \]

[Out]

(b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c - a*d)^2*Sqrt[a + b*x^4]) - (d
^(9/4)*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(4*(-c)^(3/4)*(b*c - a*d)^(5/2)) - (d
^(9/4)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(4*(-c)^(3/4)*(-(b*c) + a*d)^(5/2)
) + (b^(3/4)*(5*b*c - 11*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])/(24*a^(9/4)*(b*c - a*d)^2*Sqrt[a + b*x^4]) + (b^(1/4)*(Sqrt[b]*c - Sqrt[a]*
Sqrt[-c]*Sqrt[d])*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)*c*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + (b^(1/4)*(Sqrt[b]*c + Sqr
t[a]*Sqrt[-c]*Sqrt[d])*d^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] +
Sqrt[a]*Sqrt[d])^2*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*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2*d^2*(S
qrt[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*(b*c - a
*d)^2*(b*c + a*d)*Sqrt[a + b*x^4])

________________________________________________________________________________________

Rubi [A]  time = 1.68251, antiderivative size = 976, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {414, 527, 523, 220, 409, 1217, 1707} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{b x^4+a}}\right ) d^{9/4}}{4 (-c)^{3/4} (b c-a d)^{5/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt{b x^4+a}}\right ) d^{9/4}}{4 (-c)^{3/4} (a d-b c)^{5/2}}+\frac{\sqrt [4]{b} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \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 ) d^2}{4 \sqrt [4]{a} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{\sqrt [4]{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \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 ) d^2}{4 \sqrt [4]{a} c (b c-a 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 ) d^2}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a 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 ) d^2}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d)^2 (b c+a d) \sqrt{b x^4+a}}+\frac{b^{3/4} (5 b c-11 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 )}{24 a^{9/4} (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{b x^4+a}}+\frac{b x}{6 a (b c-a d) \left (b x^4+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)^(5/2)*(c + d*x^4)),x]

[Out]

(b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c - a*d)^2*Sqrt[a + b*x^4]) - (d
^(9/4)*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(4*(-c)^(3/4)*(b*c - a*d)^(5/2)) - (d
^(9/4)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(4*(-c)^(3/4)*(-(b*c) + a*d)^(5/2)
) + (b^(3/4)*(5*b*c - 11*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])/(24*a^(9/4)*(b*c - a*d)^2*Sqrt[a + b*x^4]) + (b^(1/4)*(Sqrt[b]*c - Sqrt[a]*
Sqrt[-c]*Sqrt[d])*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)*c*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + (b^(1/4)*(Sqrt[b]*c + Sqr
t[a]*Sqrt[-c]*Sqrt[d])*d^2*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Arc
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] +
Sqrt[a]*Sqrt[d])^2*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*(b*c - a*d)^2*(b*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2*d^2*(S
qrt[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*(b*c - a
*d)^2*(b*c + a*d)*Sqrt[a + b*x^4])

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 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 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{1}{\left (a+b x^4\right )^{5/2} \left (c+d x^4\right )} \, dx &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}-\frac{\int \frac{-5 b c+6 a d-5 b d x^4}{\left (a+b x^4\right )^{3/2} \left (c+d x^4\right )} \, dx}{6 a (b c-a d)}\\ &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{a+b x^4}}+\frac{\int \frac{5 b^2 c^2-11 a b c d+12 a^2 d^2+b d (5 b c-11 a d) x^4}{\sqrt{a+b x^4} \left (c+d x^4\right )} \, dx}{12 a^2 (b c-a d)^2}\\ &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{a+b x^4}}+\frac{d^2 \int \frac{1}{\sqrt{a+b x^4} \left (c+d x^4\right )} \, dx}{(b c-a d)^2}+\frac{(b (5 b c-11 a d)) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{12 a^2 (b c-a d)^2}\\ &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{a+b x^4}}+\frac{b^{3/4} (5 b c-11 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 )}{24 a^{9/4} (b c-a d)^2 \sqrt{a+b x^4}}+\frac{d^2 \int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c (b c-a d)^2}+\frac{d^2 \int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-c}}\right ) \sqrt{a+b x^4}} \, dx}{2 c (b c-a d)^2}\\ &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{a+b x^4}}+\frac{b^{3/4} (5 b c-11 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 )}{24 a^{9/4} (b c-a d)^2 \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) d^2\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 (b c-a d)^2 (b c+a d)}+\frac{\left (\sqrt{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d^2\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 c (b c-a d)^2 (b c+a d)}-\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d^{5/2}\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 (b c-a d)^2 (b c+a d)}+\frac{\left (\sqrt{a} \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d^{5/2}\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 (b c-a d)^2 (b c+a d)}\\ &=\frac{b x}{6 a (b c-a d) \left (a+b x^4\right )^{3/2}}+\frac{b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt{a+b x^4}}-\frac{d^{9/4} \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} (b c-a d)^{5/2}}-\frac{d^{9/4} \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} (-b c+a d)^{5/2}}+\frac{b^{3/4} (5 b c-11 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 )}{24 a^{9/4} (b c-a d)^2 \sqrt{a+b x^4}}+\frac{\sqrt [4]{b} \left (\sqrt{b}+\frac{\sqrt{a} \sqrt{d}}{\sqrt{-c}}\right ) 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} (b c-a d)^2 (b c+a d) \sqrt{a+b x^4}}+\frac{\sqrt [4]{b} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) 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} c (b c-a d)^2 (b c+a d) \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2 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 (b c-a d)^2 (b c+a d) \sqrt{a+b x^4}}+\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2 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 (b c-a d)^2 (b c+a d) \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.77853, size = 429, normalized size = 0.44 \[ -\frac{x \left (\frac{5 \left (5 a c \left (-a^2 b d \left (24 c+d x^4\right )+12 a^3 d^2+a b^2 \left (12 c^2-15 c d x^4-11 d^2 x^8\right )+5 b^3 c x^4 \left (2 c+d x^4\right )\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+2 b x^4 \left (c+d x^4\right ) \left (13 a^2 d+a b \left (11 d x^4-7 c\right )-5 b^2 c x^4\right ) \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )\right )}{\left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+\frac{b d x^4 \sqrt{\frac{b x^4}{a}+1} (11 a d-5 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}\right )}{60 a^2 \sqrt{a+b x^4} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^4)^(5/2)*(c + d*x^4)),x]

[Out]

-(x*((b*d*(-5*b*c + 11*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*(12*a^3*d^2 + 5*b^3*c*x^4*(2*c + d*x^4) - a^2*b*d*(24*c + d*x^4) + a*b^2*(12*c^2 - 15*c*d*x^4 - 11
*d^2*x^8))*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 2*b*x^4*(c + d*x^4)*(13*a^2*d - 5*b^2*c*x^
4 + a*b*(-7*c + 11*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)])))/((a + b*x^4)*(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)])))))/(60*a^2*(b*c - a*d)^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.03, size = 371, normalized size = 0.4 \begin{align*} -{\frac{x}{6\,ab \left ( ad-bc \right ) }\sqrt{b{x}^{4}+a} \left ({x}^{4}+{\frac{a}{b}} \right ) ^{-2}}-{\frac{bx \left ( 11\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{b \left ( 11\,ad-5\,bc \right ) }{12\,{a}^{2} \left ( ad-bc \right ) ^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{d}{8}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{1}{ \left ( ad-bc \right ) ^{2}{{\it \_alpha}}^{3}} \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(5/2)/(d*x^4+c),x)

[Out]

-1/6*x/a/b/(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+1/b*a)^2-1/12*b*x/a^2*(11*a*d-5*b*c)/(a*d-b*c)^2/((x^4+1/b*a)*b)^(1/
2)-1/12*b/a^2*(11*a*d-5*b*c)/(a*d-b*c)^2/(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*sum(1/(a*d-b*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)*El
lipticPi(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/2)/(d*x^4+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/2)*(d*x^4 + c)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/2)/(d*x^4+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{4}\right )^{\frac{5}{2}} \left (c + d x^{4}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(5/2)/(d*x**4+c),x)

[Out]

Integral(1/((a + b*x**4)**(5/2)*(c + d*x**4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(5/2)/(d*x^4+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/2)*(d*x^4 + c)), x)