3.188 \(\int \frac {1}{\sqrt {a-b x^4} (c-d x^4)^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt {a-b x^4} (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (5 b c-3 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (5 b c-3 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)} \]

[Out]

-1/4*d*x*(-b*x^4+a)^(1/2)/c/(-a*d+b*c)/(-d*x^4+c)-1/4*a^(1/4)*b^(3/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*(1-b*x^4/
a)^(1/2)/c/(-a*d+b*c)/(-b*x^4+a)^(1/2)+1/8*a^(1/4)*(-3*a*d+5*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)/(-b*x^4+a)^(1/2)+1/8*a^(1/4)*(-3*a*d+5*b*c)*Elli
pticPi(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)/(-b*x^4+a
)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {414, 523, 224, 221, 409, 1219, 1218} \[ -\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt {a-b x^4} (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (5 b c-3 a d) \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (5 b c-3 a d) \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)}-\frac {d x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*x*Sqrt[a - b*x^4])/(4*c*(b*c - a*d)*(c - d*x^4)) - (a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(
b^(1/4)*x)/a^(1/4)], -1])/(4*c*(b*c - a*d)*Sqrt[a - b*x^4]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ell
ipticPi[-((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)*S
qrt[a - b*x^4]) + (a^(1/4)*(5*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)*Sqrt[a - b*x^4])

Rule 221

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 224

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 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 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 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 1218

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

Rule 1219

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

________________________________________________________________________________________

Mathematica [C]  time = 0.28, size = 386, normalized size = 1.25 \[ \frac {5 a c x F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right ) \left (b d x^4 \sqrt {1-\frac {b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )-5 c \left (-4 a d+4 b c+b d x^4\right )\right )+2 d x^5 \left (b x^4 \sqrt {1-\frac {b x^4}{a}} \left (d x^4-c\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+5 c \left (a-b x^4\right )\right ) \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{20 c^2 \sqrt {a-b x^4} \left (d x^4-c\right ) (b c-a d) \left (2 x^4 \left (2 a d F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+5 a c F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c]*(-5*c*(4*b*c - 4*a*d + b*d*x^4) + b*d*x^4*Sqrt[1 - (
b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]) + 2*d*x^5*(5*c*(a - b*x^4) + b*x^4*Sq
rt[1 - (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])*(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]))/(20*c^2*(b*c - a*d)*Sqrt[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])))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.34, size = 322, normalized size = 1.04 \[ \frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{4 \left (a d -b c \right ) \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\sqrt {-b \,x^{4}+a}\, d x}{4 \left (a d -b c \right ) \left (d \,x^{4}-c \right ) c}-\frac {\left (3 a d -5 b c \right ) \left (-\frac {2 \sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3} d \EllipticPi \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , \frac {\RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} \sqrt {a}\, d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, c}-\frac {\arctanh \left (\frac {-2 \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{2} b \,x^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}\right )}{32 c d \left (a d -b c \right ) \RootOf \left (d \,\textit {\_Z}^{4}-c \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d/(a*d-b*c)/c*x*(-b*x^4+a)^(1/2)/(d*x^4-c)+1/4*b/(a*d-b*c)/c/(1/a^(1/2)*b^(1/2))^(1/2)*(-1/a^(1/2)*b^(1/2
)*x^2+1)^(1/2)*(1/a^(1/2)*b^(1/2)*x^2+1)^(1/2)/(-b*x^4+a)^(1/2)*EllipticF((1/a^(1/2)*b^(1/2))^(1/2)*x,I)-1/32/
c/d*sum((3*a*d-5*b*c)/(a*d-b*c)/_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/(1/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(-1/a^(1/2)*b^(1/2)*x^2+1)^(1/2)*(1/a^(1
/2)*b^(1/2)*x^2+1)^(1/2)/(-b*x^4+a)^(1/2)*EllipticPi((1/a^(1/2)*b^(1/2))^(1/2)*x,_alpha^2*a^(1/2)/b^(1/2)/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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a - b x^{4}} \left (- c + d x^{4}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________