Optimal. Leaf size=266 \[ -\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.506791, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/(a + b*x^4)^3,x]
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Rubi in Sympy [A] time = 86.4273, size = 257, normalized size = 0.97 \[ \frac{x \left (c + d x\right )}{8 a \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 c + 6 d x\right )}{32 a^{2} \left (a + b x^{4}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} - \frac{21 \sqrt{2} c \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{b}} + \frac{21 \sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{256 a^{\frac{11}{4}} \sqrt [4]{b}} - \frac{21 \sqrt{2} c \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{b}} + \frac{21 \sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} \sqrt [4]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/(b*x**4+a)**3,x)
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Mathematica [A] time = 0.410365, size = 249, normalized size = 0.94 \[ \frac{\frac{32 a^{7/4} x (c+d x)}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} x (7 c+6 d x)}{a+b x^4}-\frac{6 \left (8 \sqrt [4]{a} d+7 \sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{b}}+\frac{6 \left (7 \sqrt{2} \sqrt [4]{b} c-8 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{b}}-\frac{21 \sqrt{2} c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}+\frac{21 \sqrt{2} c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}}{256 a^{11/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/(a + b*x^4)^3,x]
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Maple [A] time = 0.008, size = 222, normalized size = 0.8 \[{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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((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((d*x + c)/(b*x^4 + a)^3,x, algorithm="fricas")
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Sympy [A] time = 4.26566, size = 192, normalized size = 0.72 \[ \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{2} + 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} + 194481 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{9} b d^{2} - 9633792 t^{2} a^{6} b c^{2} d - 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} - 453789 b c^{5}} \right )} \right )\right )} + \frac{11 a c x + 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/(b*x**4+a)**3,x)
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GIAC/XCAS [A] time = 0.221637, size = 346, normalized size = 1.3 \[ \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} - \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{6 \, b d x^{6} + 7 \, b c x^{5} + 10 \, a d x^{2} + 11 \, a c x}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/(b*x^4 + a)^3,x, algorithm="giac")
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