Optimal. Leaf size=932 \[ \text{result too large to display} \]
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Rubi [A] time = 1.76572, antiderivative size = 932, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c d \sqrt{\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{\frac{\sqrt{-c} \left (b-\frac{a d}{c}\right )}{\sqrt{d}}} x}{\sqrt{b x^4+a}}\right )}{4 c d \sqrt{-\frac{b c-a d}{\sqrt{-c} \sqrt{d}}}}-\frac{\sqrt [4]{b} (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (\sqrt{b} c-\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \sqrt{b x^4+a}}-\frac{\sqrt [4]{b} (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \left (\sqrt{b} c+\sqrt{a} \sqrt{-c} \sqrt{d}\right ) d \sqrt{b x^4+a}}+\frac{b^{3/4} \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} d \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) d \sqrt{b x^4+a}}+\frac{\left (\sqrt{b} \sqrt{-c}-\sqrt{a} \sqrt{d}\right ) (b c-a d) \left (\sqrt{b} x^2+\sqrt{a}\right ) \sqrt{\frac{b x^4+a}{\left (\sqrt{b} x^2+\sqrt{a}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right )^2}{4 \sqrt{a} \sqrt{b} \sqrt{-c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt{b} \sqrt{-c}+\sqrt{a} \sqrt{d}\right ) d \sqrt{b x^4+a}} \]
Warning: Unable to verify antiderivative.
[In] Int[Sqrt[a + b*x^4]/(c + d*x^4),x]
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Rubi in Sympy [A] time = 149.324, size = 814, normalized size = 0.87 \[ \text{result too large to display} \]
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]
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Mathematica [C] time = 0.237976, size = 161, normalized size = 0.17 \[ \frac{5 a c x \sqrt{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (2 x^4 \left (b c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 a d F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[a + b*x^4]/(c + d*x^4),x]
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Maple [C] time = 0.023, size = 273, normalized size = 0.3 \[{\frac{b}{d}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{1}{8\,{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}d+c \right ) }{\frac{-ad+bc}{{{\it \_alpha}}^{3}} \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}d}{c\sqrt{b{x}^{4}+a}}\sqrt{1-{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{b}{x}^{2}}{\sqrt{a}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}},{\frac{i\sqrt{a}{{\it \_alpha}}^{2}d}{c\sqrt{b}}},{1\sqrt{{\frac{-i\sqrt{b}}{\sqrt{a}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{b}}{\sqrt{a}}}}}}} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(1/2)/(d*x^4+c),x)
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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]
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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")
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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]
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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")
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