Optimal. Leaf size=127 \[ \frac{c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}} \]
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Rubi [A] time = 0.156468, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/Sqrt[-a - b*x^4],x]
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Rubi in Sympy [A] time = 14.8655, size = 112, normalized size = 0.88 \[ \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{- a - b x^{4}}} \right )}}{2 \sqrt{b}} + \frac{c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{- a - b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/(-b*x**4-a)**(1/2),x)
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Mathematica [C] time = 0.268889, size = 113, normalized size = 0.89 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}}-\frac{i c \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{-a-b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/Sqrt[-a - b*x^4],x]
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Maple [C] time = 0.023, size = 101, normalized size = 0.8 \[{c\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{-i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{-i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}-a}}}}+{\frac{d}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}-a}}}} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)/(-b*x^4-a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x + c}{\sqrt{-b x^{4} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/sqrt(-b*x^4 - a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x + c}{\sqrt{-b x^{4} - a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/sqrt(-b*x^4 - a),x, algorithm="fricas")
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Sympy [A] time = 2.92057, size = 66, normalized size = 0.52 \[ - \frac{i d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} - \frac{i c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/(-b*x**4-a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x + c}{\sqrt{-b x^{4} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/sqrt(-b*x^4 - a),x, algorithm="giac")
[Out]