Optimal. Leaf size=413 \[ \frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}+\frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}-\frac{e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \sqrt{a e^4+c d^4}} \]
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Rubi [A] time = 0.602332, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{\sqrt [4]{c} d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \Pi \left (\frac{\left (\sqrt{c} d^2+\sqrt{a} e^2\right )^2}{4 \sqrt{a} \sqrt{c} d^2 e^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a+c x^4} \left (\sqrt{a} e^2+\sqrt{c} d^2\right )}+\frac{\tan ^{-1}\left (\frac{x \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}{\sqrt{a+c x^4}}\right )}{2 d \sqrt{-\frac{a e^4+c d^4}{d^2 e^2}}}-\frac{e \tanh ^{-1}\left (\frac{a e^2+c d^2 x^2}{\sqrt{a+c x^4} \sqrt{a e^4+c d^4}}\right )}{2 \sqrt{a e^4+c d^4}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*Sqrt[a + c*x^4]),x]
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Rubi in Sympy [A] time = 56.3311, size = 367, normalized size = 0.89 \[ \frac{e \operatorname{atanh}{\left (\frac{- a e^{2} - c d^{2} x^{2}}{\sqrt{a + c x^{4}} \sqrt{a e^{4} + c d^{4}}} \right )}}{2 \sqrt{a e^{4} + c d^{4}}} + \frac{\operatorname{atan}{\left (\frac{x \sqrt{- \frac{a e^{2}}{d^{2}} - \frac{c d^{2}}{e^{2}}}}{\sqrt{a + c x^{4}}} \right )}}{2 d \sqrt{- \frac{a e^{2}}{d^{2}} - \frac{c d^{2}}{e^{2}}}} + \frac{\sqrt [4]{c} d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt{a + c x^{4}} \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )} + \frac{\sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \Pi \left (\frac{\left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )^{2}}{4 \sqrt{a} \sqrt{c} d^{2} e^{2}}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \sqrt{a + c x^{4}} \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*x**4+a)**(1/2),x)
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Mathematica [C] time = 0.433786, size = 200, normalized size = 0.48 \[ \frac{\sqrt{\frac{c x^4}{a}+1} \left (\sqrt [4]{c} d \log \left (\frac{e^2 x^2-d^2}{a e^2 \left (\sqrt{\frac{c x^4}{a}+1} \sqrt{\frac{c d^4}{a e^4}+1}+1\right )+c d^2 x^2}\right )-2 \sqrt [4]{-1} \sqrt [4]{a} e \sqrt{\frac{c d^4}{a e^4}+1} \Pi \left (\frac{i \sqrt{a} e^2}{\sqrt{c} d^2};\left .\sin ^{-1}\left (\frac{(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )\right )}{2 \sqrt [4]{c} d e \sqrt{a+c x^4} \sqrt{\frac{c d^4}{a e^4}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*Sqrt[a + c*x^4]),x]
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Maple [C] time = 0.008, size = 169, normalized size = 0.4 \[{\frac{1}{e} \left ( -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,{\frac{c{d}^{2}{x}^{2}}{{e}^{2}}}+2\,a \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{{\frac{c{d}^{4}}{{e}^{4}}}+a}}}}+{\frac{e}{d}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{-i{e}^{2}}{{d}^{2}}\sqrt{a}{\frac{1}{\sqrt{c}}}},{1\sqrt{{-i\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*x^4+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + a)*(e*x + d)),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/(sqrt(c*x^4 + a)*(e*x + d)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{4}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(c*x**4+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + a}{\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + a)*(e*x + d)),x, algorithm="giac")
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