Optimal. Leaf size=137 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.397995, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[1/((c + a/x^2 + b/x)*x^6),x]
[Out]
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Rubi in Sympy [A] time = 62.5124, size = 129, normalized size = 0.94 \[ - \frac{1}{3 a x^{3}} + \frac{b}{2 a^{2} x^{2}} - \frac{- a c + b^{2}}{a^{3} x} - \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (x \right )}}{a^{4}} + \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{4}} - \frac{\left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{4} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+a/x**2+b/x)/x**6,x)
[Out]
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Mathematica [A] time = 0.17301, size = 131, normalized size = 0.96 \[ \frac{-\frac{2 a^3}{x^3}+\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3 a^2 b}{x^2}-6 \log (x) \left (b^3-2 a b c\right )+3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))+\frac{6 a \left (a c-b^2\right )}{x}}{6 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + a/x^2 + b/x)*x^6),x]
[Out]
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Maple [A] time = 0.013, size = 214, normalized size = 1.6 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{c}{{a}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}x}}+2\,{\frac{b\ln \left ( x \right ) c}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) b}{{a}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{a}^{4}}}+2\,{\frac{{c}^{2}}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}c}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+a/x^2+b/x)/x^6,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(1/((c + b/x + a/x^2)*x^6),x, algorithm="maxima")
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Fricas [A] time = 0.331939, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (3 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (x\right ) + 3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} a^{4} x^{3}}, \frac{6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (3 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (x\right ) + 3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 28.619, size = 2105, normalized size = 15.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+a/x**2+b/x)/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.269459, size = 184, normalized size = 1.34 \[ \frac{{\left (b^{3} - 2 \, a b c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac{{\left (b^{3} - 2 \, a b c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{4}} + \frac{3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^6),x, algorithm="giac")
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