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

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]

-(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]*EllipticPi[-((Sqrt[a]*S
qrt[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]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ellipti
cPi[(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])

_______________________________________________________________________________________

Rubi [A]  time = 0.688348, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (5 b c-3 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)}-\frac{d x \sqrt{a-b x^4}}{4 c \left (c-d x^4\right ) (b c-a d)} \]

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]*EllipticPi[-((Sqrt[a]*S
qrt[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]) + (a^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (b*x^4)/a]*Ellipti
cPi[(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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 109.539, size = 270, normalized size = 0.87 \[ \frac{\sqrt [4]{a} b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{4 c \sqrt{a - b x^{4}} \left (a d - b c\right )} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (3 a d - 5 b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \sqrt{a - b x^{4}} \left (a d - b c\right )} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (3 a d - 5 b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \sqrt{a - b x^{4}} \left (a d - b c\right )} + \frac{d x \sqrt{a - b x^{4}}}{4 c \left (c - d x^{4}\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**4+a)**(1/2)/(-d*x**4+c)**2,x)

[Out]

a**(1/4)*b**(3/4)*sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(
4*c*sqrt(a - b*x**4)*(a*d - b*c)) + a**(1/4)*sqrt(1 - b*x**4/a)*(3*a*d - 5*b*c)*
elliptic_pi(-sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1/4)), -1)/(
8*b**(1/4)*c**2*sqrt(a - b*x**4)*(a*d - b*c)) + a**(1/4)*sqrt(1 - b*x**4/a)*(3*a
*d - 5*b*c)*elliptic_pi(sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1
/4)), -1)/(8*b**(1/4)*c**2*sqrt(a - b*x**4)*(a*d - b*c)) + d*x*sqrt(a - b*x**4)/
(4*c*(c - d*x**4)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [C]  time = 0.991384, size = 349, normalized size = 1.13 \[ \frac{x \left (\frac{9 a b d x^4 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{(a d-b c) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+\frac{25 a (3 a d-4 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{(b c-a d) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}+\frac{5 d \left (a-b x^4\right )}{c (b c-a d)}\right )}{20 \sqrt{a-b x^4} \left (d x^4-c\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((5*d*(a - b*x^4))/(c*(b*c - a*d)) + (25*a*(-4*b*c + 3*a*d)*AppellF1[1/4, 1/2
, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/((b*c - a*d)*(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]))) + (9*a*b*d*x^4*A
ppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c])/((-(b*c) + a*d)*(9*a*c*AppellF1
[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] + 2*x^4*(2*a*d*AppellF1[9/4, 1/2, 2, 13
/4, (b*x^4)/a, (d*x^4)/c] + b*c*AppellF1[9/4, 3/2, 1, 13/4, (b*x^4)/a, (d*x^4)/c
])))))/(20*Sqrt[a - b*x^4]*(-c + d*x^4))

_______________________________________________________________________________________

Maple [C]  time = 0.035, size = 322, normalized size = 1. \[ -{\frac{dx}{ \left ( 4\,ad-4\,bc \right ) c \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{b}{ \left ( 4\,ad-4\,bc \right ) c}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{1}{32\,cd}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{3\,ad-5\,bc}{ \left ( ad-bc \right ){{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

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-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^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/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-b^
(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*Elliptic
Pi(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))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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