3.78 \(\int \frac{1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx\)

Optimal. Leaf size=281 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{3/4} \sqrt{a-b x^4} (b c-a d)}+\frac{b x}{2 a \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)} \]

[Out]

(b*x)/(2*a*(b*c - a*d)*Sqrt[a - b*x^4]) + (b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF
[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*(b*c - a*d)*Sqrt[a - b*x^4]) - (a^
(1/4)*d*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), A
rcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)*Sqrt[a - b*x^4]) - (a^
(1/4)*d*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcS
in[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.609153, antiderivative size = 281, 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 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 a^{3/4} \sqrt{a-b x^4} (b c-a d)}+\frac{b x}{2 a \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)}-\frac{\sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt{a-b x^4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a - b*x^4)^(3/2)*(c - d*x^4)),x]

[Out]

(b*x)/(2*a*(b*c - a*d)*Sqrt[a - b*x^4]) + (b^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticF
[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*(b*c - a*d)*Sqrt[a - b*x^4]) - (a^
(1/4)*d*Sqrt[1 - (b*x^4)/a]*EllipticPi[-((Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c])), A
rcSin[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)*Sqrt[a - b*x^4]) - (a^
(1/4)*d*Sqrt[1 - (b*x^4)/a]*EllipticPi[(Sqrt[a]*Sqrt[d])/(Sqrt[b]*Sqrt[c]), ArcS
in[(b^(1/4)*x)/a^(1/4)], -1])/(2*b^(1/4)*c*(b*c - a*d)*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 106.733, size = 241, normalized size = 0.86 \[ \frac{\sqrt [4]{a} d \sqrt{1 - \frac{b x^{4}}{a}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \sqrt{a - b x^{4}} \left (a d - b c\right )} + \frac{\sqrt [4]{a} d \sqrt{1 - \frac{b x^{4}}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{b} c \sqrt{a - b x^{4}} \left (a d - b c\right )} - \frac{b x}{2 a \sqrt{a - b x^{4}} \left (a d - b c\right )} - \frac{b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 a^{\frac{3}{4}} \sqrt{a - b x^{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**4+a)**(3/2)/(-d*x**4+c),x)

[Out]

a**(1/4)*d*sqrt(1 - b*x**4/a)*elliptic_pi(-sqrt(a)*sqrt(d)/(sqrt(b)*sqrt(c)), as
in(b**(1/4)*x/a**(1/4)), -1)/(2*b**(1/4)*c*sqrt(a - b*x**4)*(a*d - b*c)) + a**(1
/4)*d*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*d - b*c)) - b*x/(2*a*sq
rt(a - b*x**4)*(a*d - b*c)) - b**(3/4)*sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1/
4)*x/a**(1/4)), -1)/(2*a**(3/4)*sqrt(a - b*x**4)*(a*d - b*c))

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Mathematica [C]  time = 0.603338, size = 329, normalized size = 1.17 \[ \frac{x \left (\frac{9 b c d x^4 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}-\frac{25 c (b c-2 a d) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{\left (c-d x^4\right ) \left (2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}-\frac{5 b}{a}\right )}{10 \sqrt{a-b x^4} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a - b*x^4)^(3/2)*(c - d*x^4)),x]

[Out]

(x*((-5*b)/a - (25*c*(b*c - 2*a*d)*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, 3/
2, 1, 9/4, (b*x^4)/a, (d*x^4)/c]))) + (9*b*c*d*x^4*AppellF1[5/4, 1/2, 1, 9/4, (b
*x^4)/a, (d*x^4)/c])/((c - d*x^4)*(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])))))/(10*(-(b*c) + a*d)*Sqrt[
a - b*x^4])

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Maple [C]  time = 0.043, size = 301, normalized size = 1.1 \[ -{\frac{bx}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{- \left ({x}^{4}-{\frac{a}{b}} \right ) b}}}}-{\frac{b}{2\,a \left ( ad-bc \right ) }\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{1}{8}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{1}{ \left ( ad-bc \right ){{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^4+a)^(3/2)/(-d*x^4+c),x)

[Out]

-1/2*b/a*x/(a*d-b*c)/(-(x^4-a/b)*b)^(1/2)-1/2*b/a/(a*d-b*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)-1/8*sum(1/(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)*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((-b*x^4 + a)^(3/2)*(d*x^4 - c)),x, algorithm="maxima")

[Out]

-integrate(1/((-b*x^4 + a)^(3/2)*(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/((-b*x^4 + a)^(3/2)*(d*x^4 - c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- a c \sqrt{a - b x^{4}} + a d x^{4} \sqrt{a - b x^{4}} + b c x^{4} \sqrt{a - b x^{4}} - b d x^{8} \sqrt{a - b x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**4+a)**(3/2)/(-d*x**4+c),x)

[Out]

-Integral(1/(-a*c*sqrt(a - b*x**4) + a*d*x**4*sqrt(a - b*x**4) + b*c*x**4*sqrt(a
- b*x**4) - b*d*x**8*sqrt(a - b*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((-b*x^4 + a)^(3/2)*(d*x^4 - c)),x, algorithm="giac")

[Out]

integrate(-1/((-b*x^4 + a)^(3/2)*(d*x^4 - c)), x)