Optimal. Leaf size=172 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
[Out]
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Rubi [A] time = 0.34306, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 63.3251, size = 158, normalized size = 0.92 \[ \frac{x \left (a g + b c + b d x + b e x^{2} + b f x^{3}\right )}{4 a b \left (a - b x^{4}\right )} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} - \frac{\left (- \sqrt{a} \sqrt{b} e + a g - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} - \frac{\left (\sqrt{a} \sqrt{b} e + a g - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 1.02567, size = 221, normalized size = 1.28 \[ \frac{\frac{4 a^{3/4} \sqrt [4]{b} (a (f+g x)+b x (c+x (d+e x)))}{a-b x^4}-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 \sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-2 \sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right )+2 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g-3 b c\right )}{16 a^{7/4} b^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]
[Out]
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Maple [B] time = 0.013, size = 305, normalized size = 1.8 \[{\frac{1}{b{x}^{4}-a} \left ( -{\frac{e{x}^{3}}{4\,a}}-{\frac{d{x}^{2}}{4\,a}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{f}{4\,b}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{bd}{8}\ln \left ({1 \left ( -{a}^{2}b+{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) \left ( -{a}^{2}b-{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,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((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,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((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223102, size = 528, normalized size = 3.07 \[ -\frac{b x^{3} e + b d x^{2} + b c x + a g x + a f}{4 \,{\left (b x^{4} - a\right )} a b} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g + \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g + \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,x, algorithm="giac")
[Out]