Optimal. Leaf size=41 \[ \frac{d \log \left (a+b x^2\right )}{2 b^2}-\frac{b c-a d}{2 b^2 \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.0879048, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{d \log \left (a+b x^2\right )}{2 b^2}-\frac{b c-a d}{2 b^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 14.0521, size = 32, normalized size = 0.78 \[ \frac{d \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{a d - b c}{2 b^{2} \left (a + b x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.01979, size = 41, normalized size = 1. \[ \frac{a d-b c}{2 b^2 \left (a+b x^2\right )}+\frac{d \log \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.013, size = 47, normalized size = 1.2 \[{\frac{d\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}+{\frac{ad}{ \left ( 2\,b{x}^{2}+2\,a \right ){b}^{2}}}-{\frac{c}{2\,b \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.3458, size = 54, normalized size = 1.32 \[ -\frac{b c - a d}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{d \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22004, size = 61, normalized size = 1.49 \[ -\frac{b c - a d -{\left (b d x^{2} + a d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.03878, size = 36, normalized size = 0.88 \[ \frac{a d - b c}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{d \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.233476, size = 88, normalized size = 2.15 \[ -\frac{d{\left (\frac{{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{2} + a\right )} b}\right )}}{2 \, b} - \frac{c}{2 \,{\left (b x^{2} + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x/(b*x^2 + a)^2,x, algorithm="giac")
[Out]