Optimal. Leaf size=41 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{c}{2 a \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.0658263, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{c}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a*c + b*c*x^2)/(x*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 12.5625, size = 37, normalized size = 0.9 \[ \frac{c}{2 a \left (a + b x^{2}\right )} + \frac{c \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{c \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*c*x**2+a*c)/x/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0219611, size = 34, normalized size = 0.83 \[ \frac{c \left (\frac{a}{a+b x^2}-\log \left (a+b x^2\right )+2 \log (x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + b*c*x^2)/(x*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.011, size = 38, normalized size = 0.9 \[{\frac{c}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{c\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*c*x^2+a*c)/x/(b*x^2+a)^3,x)
[Out]
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Maxima [A] time = 1.3293, size = 54, normalized size = 1.32 \[ \frac{c}{2 \,{\left (a b x^{2} + a^{2}\right )}} - \frac{c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{c \log \left (x^{2}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226854, size = 73, normalized size = 1.78 \[ \frac{a c -{\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b c x^{2} + a c\right )} \log \left (x\right )}{2 \,{\left (a^{2} b x^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.72782, size = 36, normalized size = 0.88 \[ c \left (\frac{1}{2 a^{2} + 2 a b x^{2}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x**2+a*c)/x/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.2273, size = 69, normalized size = 1.68 \[ \frac{c{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b c x^{2} + 2 \, a c}{2 \,{\left (b x^{2} + a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x),x, algorithm="giac")
[Out]