Optimal. Leaf size=341 \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.692528, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (21 \sqrt{b} c-5 \sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{3/4}}-\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{\left (5 \sqrt{a} e+21 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} b^{3/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac{x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2)/(a + b*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 103.555, size = 326, normalized size = 0.96 \[ \frac{x \left (c + d x + e x^{2}\right )}{8 a \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 c + 6 d x + 5 e x^{2}\right )}{32 a^{2} \left (a + b x^{4}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} + \frac{\sqrt{2} \left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e - 21 \sqrt{b} c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} e + 21 \sqrt{b} c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} e + 21 \sqrt{b} c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d*x+c)/(b*x**4+a)**3,x)
[Out]
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Mathematica [A] time = 0.631277, size = 337, normalized size = 0.99 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} e-21 \sqrt [4]{a} \sqrt{b} c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} \sqrt{b} c-5 a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{b^{3/4}}+\frac{32 a^2 x (c+x (d+e x))}{\left (a+b x^4\right )^2}-\frac{2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{2 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt{2} \sqrt{a} e+21 \sqrt{2} \sqrt{b} c\right )}{b^{3/4}}+\frac{8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2)/(a + b*x^4)^3,x]
[Out]
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Maple [A] time = 0.007, size = 396, normalized size = 1.2 \[{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e{x}^{3}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{5\,e{x}^{3}}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{5\,e\sqrt{2}}{256\,{a}^{2}b}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e\sqrt{2}}{128\,{a}^{2}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e\sqrt{2}}{128\,{a}^{2}b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d*x+c)/(b*x^4+a)^3,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((e*x^2 + d*x + c)/(b*x^4 + a)^3,x, algorithm="maxima")
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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((e*x^2 + d*x + c)/(b*x^4 + a)^3,x, algorithm="fricas")
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Sympy [A] time = 18.332, size = 558, normalized size = 1.64 \[ \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (6881280 a^{6} b^{2} c e + 4718592 a^{6} b^{2} d^{2}\right ) + t \left (153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) + 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} + 194481 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e + 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} - 118540800 t a^{4} b^{2} c^{3} e^{2} + 365783040 t a^{4} b^{2} c^{2} d^{2} e + 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} + 4536000 a^{2} b c d^{3} e^{2} - 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} - 275625 a^{2} b c^{2} e^{4} + 3024000 a^{2} b c d^{2} e^{3} - 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} + 85766121 b^{3} c^{6}} \right )} \right )\right )} + \frac{11 a c x + 10 a d x^{2} + 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d*x+c)/(b*x**4+a)**3,x)
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GIAC/XCAS [A] time = 0.224588, size = 454, normalized size = 1.33 \[ \frac{5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a x^{3} e + 10 \, a d x^{2} + 11 \, a c x}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d*x + c)/(b*x^4 + a)^3,x, algorithm="giac")
[Out]