Optimal. Leaf size=29 \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]
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Rubi [A] time = 0.0685896, antiderivative size = 29, 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 \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)*(A + B*x^2))/x^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{2 x^{2}} + \frac{b \int ^{x^{2}} B\, dx}{2} + \left (\frac{A b}{2} + \frac{B a}{2}\right ) \log{\left (x^{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(B*x**2+A)/x**3,x)
[Out]
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Mathematica [A] time = 0.0185427, size = 29, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*(A + B*x^2))/x^3,x]
[Out]
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Maple [A] time = 0.008, size = 26, normalized size = 0.9 \[{\frac{bB{x}^{2}}{2}}+A\ln \left ( x \right ) b+B\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(B*x^2+A)/x^3,x)
[Out]
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Maxima [A] time = 1.37265, size = 38, normalized size = 1.31 \[ \frac{1}{2} \, B b x^{2} + \frac{1}{2} \,{\left (B a + A b\right )} \log \left (x^{2}\right ) - \frac{A a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24138, size = 41, normalized size = 1.41 \[ \frac{B b x^{4} + 2 \,{\left (B a + A b\right )} x^{2} \log \left (x\right ) - A a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.3499, size = 26, normalized size = 0.9 \[ - \frac{A a}{2 x^{2}} + \frac{B b x^{2}}{2} + \left (A b + B a\right ) \log{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(B*x**2+A)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.262869, size = 57, normalized size = 1.97 \[ \frac{1}{2} \, B b x^{2} + \frac{1}{2} \,{\left (B a + A b\right )}{\rm ln}\left (x^{2}\right ) - \frac{B a x^{2} + A b x^{2} + A a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)/x^3,x, algorithm="giac")
[Out]