Optimal. Leaf size=60 \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]
[Out]
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Rubi [A] time = 0.150472, 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 \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a d - b c\right )}{2 b^{3} \left (a + b x^{2}\right )} + \frac{\int ^{x^{2}} d\, dx}{2 b^{2}} - \frac{\left (2 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**2+c)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0582615, size = 50, normalized size = 0.83 \[ \frac{\frac{a (b c-a d)}{a+b x^2}+(b c-2 a d) \log \left (a+b x^2\right )+b d x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^2))/(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.015, size = 74, normalized size = 1.2 \[{\frac{d{x}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) ad}{{b}^{3}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{{a}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{ac}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^2+c)/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.33961, size = 80, normalized size = 1.33 \[ \frac{d x^{2}}{2 \, b^{2}} + \frac{a b c - a^{2} d}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{{\left (b c - 2 \, a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216687, size = 105, normalized size = 1.75 \[ \frac{b^{2} d x^{4} + a b d x^{2} + a b c - a^{2} d +{\left (a b c - 2 \, a^{2} d +{\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.76607, size = 56, normalized size = 0.93 \[ - \frac{a^{2} d - a b c}{2 a b^{3} + 2 b^{4} x^{2}} + \frac{d x^{2}}{2 b^{2}} - \frac{\left (2 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**2+c)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.23864, size = 122, normalized size = 2.03 \[ \frac{\frac{{\left (b x^{2} + a\right )} d}{b^{2}} - \frac{{\left (b c - 2 \, a d\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} + \frac{\frac{a b^{2} c}{b x^{2} + a} - \frac{a^{2} b d}{b x^{2} + a}}{b^{3}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*x^3/(b*x^2 + a)^2,x, algorithm="giac")
[Out]