Optimal. Leaf size=104 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.325958, antiderivative size = 104, 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 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[1/((c + a/x^2 + b/x)*x^5),x]
[Out]
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Rubi in Sympy [A] time = 49.4208, size = 97, normalized size = 0.93 \[ - \frac{1}{2 a x^{2}} + \frac{b}{a^{2} x} + \frac{b \left (- 3 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} \sqrt{- 4 a c + b^{2}}} + \frac{\left (- a c + b^{2}\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (- a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+a/x**2+b/x)/x**5,x)
[Out]
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Mathematica [A] time = 0.251331, size = 102, normalized size = 0.98 \[ \frac{-\frac{a^2}{x^2}+2 \log (x) \left (b^2-a c\right )+\left (a c-b^2\right ) \log (a+x (b+c x))-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 a b}{x}}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + a/x^2 + b/x)*x^5),x]
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Maple [A] time = 0.012, size = 150, normalized size = 1.4 \[ -{\frac{1}{2\,a{x}^{2}}}-{\frac{\ln \left ( x \right ) c}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}}}+{\frac{b}{{a}^{2}x}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{2\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{a}^{3}}}+3\,{\frac{bc}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+a/x^2+b/x)/x^5,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(1/((c + b/x + a/x^2)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.308412, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} x^{2} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left ({\left (b^{2} - a c\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - a c\right )} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} a^{3} x^{2}}, -\frac{2 \,{\left (b^{3} - 3 \, a b c\right )} x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} - a c\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - a c\right )} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.8016, size = 1525, normalized size = 14.66 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+a/x**2+b/x)/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.294643, size = 142, normalized size = 1.37 \[ -\frac{{\left (b^{2} - a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac{{\left (b^{2} - a c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{3}} + \frac{2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)*x^5),x, algorithm="giac")
[Out]