Optimal. Leaf size=87 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.15086, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/(a - b*x^4),x]
[Out]
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Rubi in Sympy [A] time = 21.5933, size = 82, normalized size = 0.94 \[ \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} + \frac{c \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} \sqrt [4]{b}} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} \sqrt [4]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/(-b*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0632107, size = 134, normalized size = 1.54 \[ \frac{-\left (\sqrt [4]{a} d+\sqrt [4]{b} c\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \sqrt [4]{b} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/(a - b*x^4),x]
[Out]
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Maple [A] time = 0.008, size = 101, normalized size = 1.2 \[{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+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 + c)/(b*x^4 - a),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x + c)/(b*x^4 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.36486, size = 126, normalized size = 1.45 \[ - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b d^{2} - 16 t a b c^{2} d + a d^{4} - b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 128 t^{3} a^{3} b d^{2} + 16 t^{2} a^{2} b c^{2} d + 8 t a^{2} d^{4} - 4 t a b c^{4} + 5 a c^{2} d^{3}}{4 a c d^{4} + b c^{5}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/(-b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.216631, size = 304, normalized size = 3.49 \[ \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b} - \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b d + \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b d + \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(d*x + c)/(b*x^4 - a),x, algorithm="giac")
[Out]