Optimal. Leaf size=86 \[ \frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]
[Out]
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Rubi [A] time = 0.100966, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/(a - b*x^4),x]
[Out]
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Rubi in Sympy [A] time = 16.0188, size = 76, normalized size = 0.88 \[ - \frac{\left (\sqrt{a} d - \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} + \frac{\left (\sqrt{a} d + \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(-b*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0469434, size = 95, normalized size = 1.1 \[ \frac{2 \left (\sqrt{b} c-\sqrt{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt{a} d+\sqrt{b} c\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/(a - b*x^4),x]
[Out]
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Maple [B] time = 0.006, size = 122, normalized size = 1.4 \[{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{d}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(-b*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^2 + c)/(b*x^4 - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.331886, size = 1019, normalized size = 11.85 \[ \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x +{\left (a^{3} b^{2} d \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt{\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac{1}{4} \, \sqrt{\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x -{\left (a^{3} b^{2} d \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt{\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x +{\left (a^{3} b^{2} d \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt{-\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) + \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x -{\left (a^{3} b^{2} d \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt{-\frac{a b \sqrt{\frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^2 + c)/(b*x^4 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.99784, size = 110, normalized size = 1.28 \[ - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} - 64 t^{2} a^{2} b^{2} c d - a^{2} d^{4} + 2 a b c^{2} d^{2} - b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{3} b^{2} d + 12 t a^{2} b c d^{2} + 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/(-b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.290926, size = 347, normalized size = 4.03 \[ \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x^2 + c)/(b*x^4 - a),x, algorithm="giac")
[Out]