Optimal. Leaf size=110 \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]
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Rubi [A] time = 0.18865, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)/(a - b*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 32.9588, size = 102, normalized size = 0.93 \[ \frac{x \left (c + d x\right )}{4 a \left (a - b x^{4}\right )} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{b}} + \frac{3 c \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} \sqrt [4]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)/(-b*x**4+a)**2,x)
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Mathematica [A] time = 0.343325, size = 168, normalized size = 1.53 \[ \frac{\frac{4 a x (c+d x)}{a-b x^4}-\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c+2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c-2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{6 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{2 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{16 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)/(a - b*x^4)^2,x]
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Maple [A] time = 0.007, size = 142, normalized size = 1.3 \[ -{\frac{cx}{4\,a \left ( b{x}^{4}-a \right ) }}+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{d}{8\,a}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)/(-b*x^4+a)^2,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)^2,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)^2,x, algorithm="fricas")
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Sympy [A] time = 2.41347, size = 155, normalized size = 1.41 \[ \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{2} - 2048 t^{2} a^{4} b d^{2} + 1152 t a^{2} b c^{2} d + 16 a d^{4} - 81 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{32768 t^{3} a^{6} b d^{2} + 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} + 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} + 243 b c^{5}} \right )} \right )\right )} - \frac{c x + d x^{2}}{- 4 a^{2} + 4 a b x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)/(-b*x**4+a)**2,x)
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GIAC/XCAS [A] time = 0.221915, size = 343, normalized size = 3.12 \[ \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{d x^{2} + c x}{4 \,{\left (b x^{4} - a\right )} a} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)/(b*x^4 - a)^2,x, algorithm="giac")
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