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