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

Optimal. Leaf size=362 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (a d+2 b c)}{4 a c \sqrt{a-b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt{a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]

[Out]

(b*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*Sqrt[a - b*x^4]) - (d*x)/(4*c*(b*c - a*
d)*Sqrt[a - b*x^4]*(c - d*x^4)) + (b^(3/4)*(2*b*c + a*d)*Sqrt[1 - (b*x^4)/a]*Ell
ipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(4*a^(3/4)*c*(b*c - a*d)^2*Sqrt[a - b*x
^4]) - (3*a^(1/4)*d*(3*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])/(8*b^(1/4)*c^2*(b*c -
a*d)^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*d*(3*b*c - a*d)*Sqrt[1 - (b*x^4)/a]*Ellipt
icPi[(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)^2*Sqrt[a - b*x^4])

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Rubi [A]  time = 1.07374, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{b^{3/4} \sqrt{1-\frac{b x^4}{a}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}-\frac{3 \sqrt [4]{a} d \sqrt{1-\frac{b x^4}{a}} (3 b c-a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 \sqrt{a-b x^4} (b c-a d)^2}+\frac{b x (a d+2 b c)}{4 a c \sqrt{a-b x^4} (b c-a d)^2}-\frac{d x}{4 c \sqrt{a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(b*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*Sqrt[a - b*x^4]) - (d*x)/(4*c*(b*c - a*
d)*Sqrt[a - b*x^4]*(c - d*x^4)) + (b^(3/4)*(2*b*c + a*d)*Sqrt[1 - (b*x^4)/a]*Ell
ipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(4*a^(3/4)*c*(b*c - a*d)^2*Sqrt[a - b*x
^4]) - (3*a^(1/4)*d*(3*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])/(8*b^(1/4)*c^2*(b*c -
a*d)^2*Sqrt[a - b*x^4]) - (3*a^(1/4)*d*(3*b*c - a*d)*Sqrt[1 - (b*x^4)/a]*Ellipt
icPi[(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)^2*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 175.193, size = 323, normalized size = 0.89 \[ \frac{3 \sqrt [4]{a} d \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - 3 b c\right ) \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \sqrt{a - b x^{4}} \left (a d - b c\right )^{2}} + \frac{3 \sqrt [4]{a} d \sqrt{1 - \frac{b x^{4}}{a}} \left (a d - 3 b c\right ) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{b} \sqrt{c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{8 \sqrt [4]{b} c^{2} \sqrt{a - b x^{4}} \left (a d - b c\right )^{2}} + \frac{d x}{4 c \sqrt{a - b x^{4}} \left (c - d x^{4}\right ) \left (a d - b c\right )} + \frac{b x \left (a d + 2 b c\right )}{4 a c \sqrt{a - b x^{4}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{4}} \sqrt{1 - \frac{b x^{4}}{a}} \left (a d + 2 b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{4 a^{\frac{3}{4}} c \sqrt{a - b x^{4}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.18553, size = 465, normalized size = 1.28 \[ \frac{x \left (\frac{25 \left (3 a^2 d^2-8 a b c d+2 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}{2 x^4 \left (2 a d F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )}+\frac{9 a c \left (5 a^2 d^2-6 a b d^2 x^4+2 b^2 c \left (5 c-6 d x^4\right )\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )-10 x^4 \left (-a^2 d^2+a b d^2 x^4-2 b^2 c \left (c-d x^4\right )\right ) \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}{a c \left (2 x^4 \left (2 a d F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )+b c F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},\frac{d x^4}{c}\right )\right )}\right )}{20 \sqrt{a-b x^4} \left (c-d x^4\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((25*(2*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4, (b*x^4)/a
, (d*x^4)/c])/(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*c*(5*a^2*d^2 - 6*a*b*d^2*x^4 + 2*b^2*c*(5*
c - 6*d*x^4))*AppellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, (d*x^4)/c] - 10*x^4*(-(a^2*d
^2) + a*b*d^2*x^4 - 2*b^2*c*(c - d*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]))/(a*c
*(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*(b*c - a*d)^2*Sqrt[a - b*x^4]*(c - d*x^4))

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Maple [C]  time = 0.058, size = 375, normalized size = 1. \[ -{\frac{{d}^{2}x}{4\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{4}-c \right ) }\sqrt{-b{x}^{4}+a}}+{\frac{{b}^{2}x}{2\,a \left ( ad-bc \right ) ^{2}}{\frac{1}{\sqrt{- \left ({x}^{4}-{\frac{a}{b}} \right ) b}}}}+{1 \left ({\frac{bd}{4\, \left ( ad-bc \right ) ^{2}c}}+{\frac{{b}^{2}}{2\,a \left ( ad-bc \right ) ^{2}}} \right ) \sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}-{\frac{3}{32\,c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d-c \right ) }{\frac{ad-3\,bc}{ \left ( ad-bc \right ) ^{2}{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{-2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{-b{x}^{4}+a}}\sqrt{1-{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}},{\frac{\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{-{\frac{\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*d^2/(a*d-b*c)^2/c*x*(-b*x^4+a)^(1/2)/(d*x^4-c)+1/2*b^2/a*x/(a*d-b*c)^2/(-(x
^4-a/b)*b)^(1/2)+(1/4*d*b/(a*d-b*c)^2/c+1/2*b^2/a/(a*d-b*c)^2)/(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)-3/32/c*sum((a*d-3*b*c)/(a*d-b*c)^
2/_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. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}{\left (d x^{4} - c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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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)**(3/2)/(-d*x**4+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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