Optimal. Leaf size=318 \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}-\frac{a f-b x \left (c+d x+e x^2\right )}{4 a b \left (a+b x^4\right )} \]
[Out]
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Rubi [A] time = 0.582672, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{3/4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}-\frac{a f-b x \left (c+d x+e x^2\right )}{4 a b \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 94.1335, size = 294, normalized size = 0.92 \[ - \frac{a f - b x \left (c + d x + e x^{2}\right )}{4 a b \left (a + b x^{4}\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} + \frac{\sqrt{2} \left (\sqrt{a} e - 3 \sqrt{b} c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e - 3 \sqrt{b} c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e + 3 \sqrt{b} c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} e + 3 \sqrt{b} c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.634726, size = 315, normalized size = 0.99 \[ \frac{\sqrt{2} \sqrt [4]{b} \left (a^{3/4} e-3 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+\sqrt{2} \sqrt [4]{b} \left (3 \sqrt [4]{a} \sqrt{b} c-a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\frac{8 a (a f-b x (c+x (d+e x)))}{a+b x^4}-2 \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )+2 \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 \sqrt [4]{a} \sqrt [4]{b} d+\sqrt{2} \sqrt{a} e+3 \sqrt{2} \sqrt{b} c\right )}{32 a^2 b} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3)/(a + b*x^4)^2,x]
[Out]
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Maple [A] time = 0.011, size = 362, normalized size = 1.1 \[{\frac{cx}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{3\,c\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,c\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{d}{4\,a}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}+a \right ) }}+{\frac{e\sqrt{2}}{32\,ab}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{16\,ab}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{16\,ab}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{f{x}^{4}}{4\,a \left ( b{x}^{4}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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((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((f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 32.1885, size = 517, normalized size = 1.63 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (3072 a^{4} b^{2} c e + 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} - 1152 a^{2} b^{2} c^{2} d\right ) + a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} + 81 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} b^{2} e^{3} - 36864 t^{3} a^{6} b^{3} c^{2} e + 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e + 13824 t^{2} a^{4} b^{3} c^{3} d + 144 t a^{4} b c e^{4} + 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} + 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} + 120 a^{2} b c d^{3} e^{2} - 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} - 9 a^{2} b c^{2} e^{4} + 96 a^{2} b c d^{2} e^{3} - 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} + 729 b^{3} c^{6}} \right )} \right )\right )} + \frac{- a f + b c x + b d x^{2} + b e x^{3}}{4 a^{2} b + 4 a b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**3+e*x**2+d*x+c)/(b*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.229595, size = 427, normalized size = 1.34 \[ \frac{b x^{3} e + b d x^{2} + b c x - a f}{4 \,{\left (b x^{4} + a\right )} a b} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{a b} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^2,x, algorithm="giac")
[Out]