Optimal. Leaf size=62 \[ -\frac{(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac{b^2 x^2}{2 d^2} \]
[Out]
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Rubi [A] time = 0.150019, antiderivative size = 62, 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 \[ -\frac{(b c-a d)^2}{2 d^3 \left (c+d x^2\right )}-\frac{b (b c-a d) \log \left (c+d x^2\right )}{d^3}+\frac{b^2 x^2}{2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{d^{3}} + \frac{\int ^{x^{2}} b^{2}\, dx}{2 d^{2}} - \frac{\left (a d - b c\right )^{2}}{2 d^{3} \left (c + d x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.0759851, size = 56, normalized size = 0.9 \[ \frac{-\frac{(b c-a d)^2}{c+d x^2}+2 b (a d-b c) \log \left (c+d x^2\right )+b^2 d x^2}{2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Maple [A] time = 0.015, size = 97, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{2}}{2\,{d}^{2}}}-{\frac{{a}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{abc}{{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{b\ln \left ( d{x}^{2}+c \right ) a}{{d}^{2}}}-{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) c}{{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^2/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.33241, size = 100, normalized size = 1.61 \[ \frac{b^{2} x^{2}}{2 \, d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (d^{4} x^{2} + c d^{3}\right )}} - \frac{{\left (b^{2} c - a b d\right )} \log \left (d x^{2} + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221388, size = 136, normalized size = 2.19 \[ \frac{b^{2} d^{2} x^{4} + b^{2} c d x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (d^{4} x^{2} + c d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.6142, size = 68, normalized size = 1.1 \[ \frac{b^{2} x^{2}}{2 d^{2}} + \frac{b \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c d^{3} + 2 d^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.236121, size = 149, normalized size = 2.4 \[ \frac{{\left (d x^{2} + c\right )} b^{2}}{2 \, d^{3}} + \frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left (\frac{{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} - \frac{\frac{b^{2} c^{2} d}{d x^{2} + c} - \frac{2 \, a b c d^{2}}{d x^{2} + c} + \frac{a^{2} d^{3}}{d x^{2} + c}}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^2,x, algorithm="giac")
[Out]