Optimal. Leaf size=53 \[ \frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{b c}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]
[Out]
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Rubi [A] time = 0.088698, antiderivative size = 53, 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 \[ \frac{b c \log \left (a+b x^2\right )}{a^3}-\frac{2 b c \log (x)}{a^3}-\frac{b c}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 15.6278, size = 51, normalized size = 0.96 \[ - \frac{b c}{2 a^{2} \left (a + b x^{2}\right )} - \frac{c}{2 a^{2} x^{2}} - \frac{b c \log{\left (x^{2} \right )}}{a^{3}} + \frac{b c \log{\left (a + b x^{2} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*c*x**2+a*c)/x**3/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.0604297, size = 42, normalized size = 0.79 \[ -\frac{c \left (a \left (\frac{b}{a+b x^2}+\frac{1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + b*c*x^2)/(x^3*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.013, size = 50, normalized size = 0.9 \[ -{\frac{c}{2\,{a}^{2}{x}^{2}}}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*c*x^2+a*c)/x^3/(b*x^2+a)^3,x)
[Out]
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Maxima [A] time = 1.35243, size = 77, normalized size = 1.45 \[ -\frac{2 \, b c x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{b c \log \left (b x^{2} + a\right )}{a^{3}} - \frac{b c \log \left (x^{2}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228062, size = 108, normalized size = 2.04 \[ -\frac{2 \, a b c x^{2} + a^{2} c - 2 \,{\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} c x^{4} + a b c x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.09321, size = 51, normalized size = 0.96 \[ c \left (- \frac{a + 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x**2+a*c)/x**3/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.24299, size = 76, normalized size = 1.43 \[ -\frac{b c{\rm ln}\left (x^{2}\right )}{a^{3}} + \frac{b c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac{2 \, b c x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/((b*x^2 + a)^3*x^3),x, algorithm="giac")
[Out]