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

Optimal. Leaf size=162 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.326382, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.4592, size = 143, normalized size = 0.88 \[ \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.234785, size = 156, normalized size = 0.96 \[ -\frac{5 a c x F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\sqrt{a-b x^4} \left (d x^4-c\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a, (d*x^4)/c])/(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])))

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Maple [C]  time = 0.023, size = 183, normalized size = 1.1 \[ -{\frac{1}{8\,d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{1}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{d{{\it \_alpha}}^{3}}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/d*sum(1/_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)*El
lipticPi(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- c \sqrt{a - b x^{4}} + d x^{4} \sqrt{a - b x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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