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

Optimal. Leaf size=240 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{d \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d \sqrt{a-b x^4}} \]

[Out]

(a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])
/(d*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sq
rt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*
c*d*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - 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])/(2*b^(1/4)*c*d
*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.448504, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{\sqrt [4]{a} b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{d \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d \sqrt{a-b x^4}}-\frac{\sqrt [4]{a} \sqrt{1-\frac{b x^4}{a}} (b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(a^(1/4)*b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])
/(d*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - a*d)*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sq
rt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*
c*d*Sqrt[a - b*x^4]) - (a^(1/4)*(b*c - 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])/(2*b^(1/4)*c*d
*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 73.1836, size = 211, normalized size = 0.88 \[ \frac{\sqrt [4]{a} b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d \sqrt{a - b x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c d \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/2)/(-d*x**4+c),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)/(
d*sqrt(a - b*x**4)) + a**(1/4)*sqrt(1 - b*x**4/a)*(a*d - b*c)*elliptic_pi(-sqrt(
a)*sqrt(d)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1/4)), -1)/(2*b**(1/4)*c*d*sqr
t(a - b*x**4)) + a**(1/4)*sqrt(1 - b*x**4/a)*(a*d - b*c)*elliptic_pi(sqrt(a)*sqr
t(d)/(sqrt(b)*sqrt(c)), asin(b**(1/4)*x/a**(1/4)), -1)/(2*b**(1/4)*c*d*sqrt(a -
b*x**4))

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Mathematica [C]  time = 0.276908, size = 155, normalized size = 0.65 \[ -\frac{5 a c x \sqrt{a-b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (b c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-2 a d F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.025, size = 259, normalized size = 1.1 \[{\frac{b}{d}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}+{\frac{1}{8\,{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{-ad+bc}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b/d/(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)+1/8/d^2*sum(
(-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)*Elli
pticPi(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. \[ -\int \frac{\sqrt{-b x^{4} + a}}{d x^{4} - c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate(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(-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. \[ - \int \frac{\sqrt{a - b x^{4}}}{- c + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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