Optimal. Leaf size=60 \[ \frac{a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac{(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}+\frac{B x^2}{2 b^2} \]
[Out]
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Rubi [A] time = 0.15111, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac{(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}+\frac{B x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a \left (A b - B a\right )}{2 b^{3} \left (a + b x^{2}\right )} + \frac{\int ^{x^{2}} B\, dx}{2 b^{2}} + \frac{\left (A b - 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**2+A)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.057108, size = 50, normalized size = 0.83 \[ \frac{\frac{a (A b-a B)}{a+b x^2}+(A b-2 a B) \log \left (a+b x^2\right )+b B x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.015, size = 74, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) Ba}{{b}^{3}}}+{\frac{aA}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^2+A)/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.35179, size = 81, normalized size = 1.35 \[ \frac{B x^{2}}{2 \, b^{2}} - \frac{B a^{2} - A a b}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227237, size = 109, normalized size = 1.82 \[ \frac{B b^{2} x^{4} + B a b x^{2} - B a^{2} + A a b -{\left (2 \, B a^{2} - A a b +{\left (2 \, B a b - A b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.77839, size = 56, normalized size = 0.93 \[ \frac{B x^{2}}{2 b^{2}} - \frac{- A a b + B a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**2+A)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223952, size = 123, normalized size = 2.05 \[ \frac{\frac{{\left (b x^{2} + a\right )} B}{b^{2}} + \frac{{\left (2 \, B a - A b\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} - \frac{\frac{B a^{2} b}{b x^{2} + a} - \frac{A a b^{2}}{b x^{2} + a}}{b^{3}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^3/(b*x^2 + a)^2,x, algorithm="giac")
[Out]