Optimal. Leaf size=221 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (-a g+7 b c+6 b d x+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
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Rubi [A] time = 0.541059, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (-a g+7 b c+6 b d x+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x]
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Rubi in Sympy [A] time = 96.4441, size = 212, normalized size = 0.96 \[ \frac{x \left (a g + b c + b d x + b e x^{2} + b f x^{3}\right )}{8 a b \left (a - b x^{4}\right )^{2}} + \frac{4 a f - x \left (a g - 7 b c - 6 b d x - 5 b e x^{2}\right )}{32 a^{2} b \left (a - b x^{4}\right )} + \frac{3 d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} - \frac{\left (- 5 \sqrt{a} \sqrt{b} e + 3 a g - 21 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{\left (5 \sqrt{a} \sqrt{b} e + 3 a g - 21 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.99216, size = 263, normalized size = 1.19 \[ \frac{\frac{4 a^{3/4} \sqrt [4]{b} \left (a^2 (4 f+3 g x)+a b x \left (11 c+x \left (10 d+9 e x+g x^3\right )\right )-b^2 x^5 (7 c+x (6 d+5 e x))\right )}{\left (a-b x^4\right )^2}-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (12 \sqrt [4]{a} b^{3/4} d+5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-12 \sqrt [4]{a} b^{3/4} d+5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )+12 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{128 a^{11/4} b^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^3,x]
[Out]
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Maple [A] time = 0.02, size = 344, normalized size = 1.6 \[ -{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{2}} \left ({\frac{5\,be{x}^{7}}{32\,{a}^{2}}}+{\frac{3\,bd{x}^{6}}{16\,{a}^{2}}}-{\frac{ \left ( ag-7\,bc \right ){x}^{5}}{32\,{a}^{2}}}-{\frac{9\,e{x}^{3}}{32\,a}}-{\frac{5\,d{x}^{2}}{16\,a}}-{\frac{ \left ( 3\,ag+11\,bc \right ) x}{32\,ab}}-{\frac{f}{8\,b}} \right ) }-{\frac{3\,g}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{3\,g}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{3\,bd}{32}\ln \left ({1 \left ( -{a}^{3}b+{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) \left ( -{a}^{3}b-{x}^{2}\sqrt{{a}^{5}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{5}{b}^{3}}}}}-{\frac{5\,e}{64\,{a}^{2}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,{a}^{2}b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^3,x)
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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)^3,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)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.225514, size = 595, normalized size = 2.69 \[ -\frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{-a b} b^{2} d - 21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - a b g x^{5} - 9 \, a b x^{3} e - 10 \, a b d x^{2} - 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2} b} \]
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)^3,x, algorithm="giac")
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