\(\int \frac {1}{(a-b x^4)^{5/2} (c-d x^4)^2} \, dx\) [190]

Optimal result

Integrand size = 23, antiderivative size = 439 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {b^{3/4} \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}} \]

[Out]

1/12*b*(3*a*d+2*b*c)*x/a/c/(-a*d+b*c)^2/(-b*x^4+a)^(3/2)-1/4*d*x/c/(-a*d+b*c)/(-b*x^4+a)^(3/2)/(-d*x^4+c)+1/12
*b*(-3*a^2*d^2-17*a*b*c*d+5*b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)+1/12*b^(3/4)*(-3*a^2*d^2-17*a*b*c*d
+5*b^2*c^2)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/a)^(1/2)/a^(7/4)/c/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)+1/8*a^(1/
4)*d^2*(-3*a*d+13*b*c)*EllipticPi(b^(1/4)*x/a^(1/4),-a^(1/2)*d^(1/2)/b^(1/2)/c^(1/2),I)*(1-b*x^4/a)^(1/2)/b^(1
/4)/c^2/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)+1/8*a^(1/4)*d^2*(-3*a*d+13*b*c)*EllipticPi(b^(1/4)*x/a^(1/4),a^(1/2)*d^(
1/2)/b^(1/2)/c^(1/2),I)*(1-b*x^4/a)^(1/2)/b^(1/4)/c^2/(-a*d+b*c)^3/(-b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {425, 541, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\frac {b x \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right )}{12 a^2 c \sqrt {a-b x^4} (b c-a d)^3}+\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-17 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c \sqrt {a-b x^4} (b c-a d)^3}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} (13 b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^3}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} (13 b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^3}-\frac {d x}{4 c \left (a-b x^4\right )^{3/2} \left (c-d x^4\right ) (b c-a d)}+\frac {b x (3 a d+2 b c)}{12 a c \left (a-b x^4\right )^{3/2} (b c-a d)^2} \]

[In]

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

[Out]

(b*(2*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a - b*x^4)^(3/2)) + (b*(5*b^2*c^2 - 17*a*b*c*d - 3*a^2*d^2)*x)/(1
2*a^2*c*(b*c - a*d)^3*Sqrt[a - b*x^4]) - (d*x)/(4*c*(b*c - a*d)*(a - b*x^4)^(3/2)*(c - d*x^4)) + (b^(3/4)*(5*b
^2*c^2 - 17*a*b*c*d - 3*a^2*d^2)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(12*a^(7/4)*c
*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^(1/4)*d^2*(13*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt
[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^3*Sqrt[a - b*x^4]) + (a^
(1/4)*d^2*(13*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcSin[(b^(1/4)
*x)/a^(1/4)], -1])/(8*b^(1/4)*c^2*(b*c - a*d)^3*Sqrt[a - b*x^4])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 418

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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 537

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 541

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 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a)]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}-\frac {\int \frac {-4 b c+3 a d-9 b d x^4}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}-\frac {\int \frac {-2 \left (10 b^2 c^2-24 a b c d+9 a^2 d^2\right )+10 b d (2 b c+3 a d) x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{24 a c (b c-a d)^2} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}-\frac {\int \frac {-4 \left (5 b^3 c^3-17 a b^2 c^2 d+36 a^2 b c d^2-9 a^3 d^3\right )+4 b d \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{48 a^2 c (b c-a d)^3} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {\left (d^2 (13 b c-3 a d)\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)^3}+\frac {\left (b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{12 a^2 c (b c-a d)^3} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {\left (d^2 (13 b c-3 a d)\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^3}+\frac {\left (d^2 (13 b c-3 a d)\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^3}+\frac {\left (b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {b^{3/4} \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} c (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\left (d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\left (d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 (b c-a d)^3 \sqrt {a-b x^4}} \\ & = \frac {b (2 b c+3 a d) x}{12 a c (b c-a d)^2 \left (a-b x^4\right )^{3/2}}+\frac {b \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \left (a-b x^4\right )^{3/2} \left (c-d x^4\right )}+\frac {b^{3/4} \left (5 b^2 c^2-17 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{12 a^{7/4} c (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 (13 b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^3 \sqrt {a-b x^4}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.88 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\frac {x \left (\frac {b d \left (-5 b^2 c^2+17 a b c d+3 a^2 d^2\right ) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{a^2 c^2}+5 \left (\frac {5 b^3 c}{a^2}-\frac {17 b^2 d}{a}-\frac {2 b^2 d}{a-b x^4}+\frac {2 b^3 c}{a^2-a b x^4}-\frac {3 a d^3}{c^2-c d x^4}+\frac {3 b d^3 x^4}{c^2-c d x^4}+\frac {5 \left (5 b^3 c^3-17 a b^2 c^2 d+36 a^2 b c d^2-9 a^3 d^3\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{a \left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )\right )}{60 (b c-a d)^3 \sqrt {a-b x^4}} \]

[In]

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

[Out]

(x*((b*d*(-5*b^2*c^2 + 17*a*b*c*d + 3*a^2*d^2)*x^4*Sqrt[1 - (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (
d*x^4)/c])/(a^2*c^2) + 5*((5*b^3*c)/a^2 - (17*b^2*d)/a - (2*b^2*d)/(a - b*x^4) + (2*b^3*c)/(a^2 - a*b*x^4) - (
3*a*d^3)/(c^2 - c*d*x^4) + (3*b*d^3*x^4)/(c^2 - c*d*x^4) + (5*(5*b^3*c^3 - 17*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 9
*a^3*d^3)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/(a*(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*(b*c - a*d)^3*Sqrt[a - b*x^4])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.43 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {b \,d^{3} x \sqrt {-b \,x^{4}+a}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b d \,x^{4}-b c \right )}+\frac {x \sqrt {-b \,x^{4}+a}}{6 \left (a d -b c \right )^{2} a \left (x^{4}-\frac {a}{b}\right )^{2}}+\frac {b^{2} x \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b \,d^{2}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{2} \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -13 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\)	\(484\)
elliptic	\(-\frac {b \,d^{3} x \sqrt {-b \,x^{4}+a}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b d \,x^{4}-b c \right )}+\frac {x \sqrt {-b \,x^{4}+a}}{6 \left (a d -b c \right )^{2} a \left (x^{4}-\frac {a}{b}\right )^{2}}+\frac {b^{2} x \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b \,d^{2}}{4 \left (a d -b c \right ) c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{2} \left (17 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{3}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -13 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{3} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\)	\(484\)

[In]

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

[Out]

-1/4*b*d^3/(a*d-b*c)/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*(-b*x^4+a)^(1/2)/(b*d*x^4-b*c)+1/6/(a*d-b*c)^2/a*x*(-b*x^
4+a)^(1/2)/(x^4-a/b)^2+1/12*b^2*x/a^2*(17*a*d-5*b*c)/(a*d-b*c)^3/(-(x^4-a/b)*b)^(1/2)+(1/4*b*d^2/(a*d-b*c)/c/(
a^2*d^2-2*a*b*c*d+b^2*c^2)+1/12*b^2/a^2*(17*a*d-5*b*c)/(a*d-b*c)^3)/(1/a^(1/2)*b^(1/2))^(1/2)*(1-x^2*b^(1/2)/a
^(1/2))^(1/2)*(1+x^2*b^(1/2)/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-1/32*d/c
*sum((3*a*d-13*b*c)/(a*d-b*c)^3/_alpha^3*(-1/(1/d*(a*d-b*c))^(1/2)*arctanh(1/2*(-2*_alpha^2*b*x^2+2*a)/(1/d*(a
*d-b*c))^(1/2)/(-b*x^4+a)^(1/2))-2/(1/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(1-x^2*b^(1/2)/a^(1/2))^(1/2)*(1+x^2
*b^(1/2)/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*b^(1/2))^(1/2),a^(1/2)/b^(1/2)*_alpha^2/c*d,(
-1/a^(1/2)*b^(1/2))^(1/2)/(1/a^(1/2)*b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*d-c))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{\left (a - b x^{4}\right )^{\frac {5}{2}} \left (- c + d x^{4}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{5/2}\,{\left (c-d\,x^4\right )}^2} \,d x \]

[In]

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

[Out]

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