Optimal. Leaf size=229 \[ \frac{(A b-2 a C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a}+\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]
[Out]
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Rubi [A] time = 0.627816, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{(A b-2 a C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac{A \log (x)}{a}+\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 76.694, size = 216, normalized size = 0.94 \[ \frac{A \log{\left (x^{2} \right )}}{2 a} - \frac{A \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a} - \frac{\sqrt{2} B \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} B \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\left (A b - 2 C a\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+B*x+A)/x/(c*x**4+b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.932439, size = 285, normalized size = 1.24 \[ -\frac{\left (A \left (\sqrt{b^2-4 a c}+b\right )-2 a C\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{4 a \sqrt{b^2-4 a c}}-\frac{\left (A \left (\sqrt{b^2-4 a c}-b\right )+2 a C\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{4 a \sqrt{b^2-4 a c}}+\frac{A \log (x)}{a}+\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x]
[Out]
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Maple [B] time = 0.038, size = 488, normalized size = 2.1 \[ -4\,{\frac{c\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) A}{16\,ac-4\,{b}^{2}}}+{\frac{A{b}^{2}}{a \left ( 16\,ac-4\,{b}^{2} \right ) }\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) }-{\frac{Ab}{a \left ( 16\,ac-4\,{b}^{2} \right ) }\sqrt{-4\,ac+{b}^{2}}\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) }+2\,{\frac{\sqrt{-4\,ac+{b}^{2}}\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) C}{16\,ac-4\,{b}^{2}}}+4\,{\frac{\sqrt{-4\,ac+{b}^{2}}cB\sqrt{2}}{ \left ( 16\,ac-4\,{b}^{2} \right ) \sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}\arctan \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }-4\,{\frac{c\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) A}{16\,ac-4\,{b}^{2}}}+{\frac{A{b}^{2}}{a \left ( 16\,ac-4\,{b}^{2} \right ) }\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) }+{\frac{Ab}{a \left ( 16\,ac-4\,{b}^{2} \right ) }\sqrt{-4\,ac+{b}^{2}}\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) }-2\,{\frac{\sqrt{-4\,ac+{b}^{2}}\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) C}{16\,ac-4\,{b}^{2}}}+4\,{\frac{\sqrt{-4\,ac+{b}^{2}}cB\sqrt{2}}{ \left ( 16\,ac-4\,{b}^{2} \right ) \sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}{\it Artanh} \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }+{\frac{A\ln \left ( x \right ) }{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)/x/(c*x^4+b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{A \log \left (x\right )}{a} - \frac{\int \frac{A c x^{3} - B a -{\left (C a - A b\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)*x),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)*x),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((C*x**2+B*x+A)/x/(c*x**4+b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.866925, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)*x),x, algorithm="giac")
[Out]