Optimal. Leaf size=67 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]
[Out]
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Rubi [A] time = 0.151324, antiderivative size = 67, 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 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 25.5464, size = 56, normalized size = 0.84 \[ \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{b \left (a d - b c\right )}{d^{3} \left (c + d x^{2}\right )} - \frac{\left (a d - b c\right )^{2}}{4 d^{3} \left (c + d x^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.0503628, size = 75, normalized size = 1.12 \[ \frac{-a^2 d^2-2 a b d \left (c+2 d x^2\right )+b^2 c \left (3 c+4 d x^2\right )+2 b^2 \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 d^3 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x^2)^2)/(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.014, size = 105, normalized size = 1.6 \[ -{\frac{ab}{{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{{d}^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{a}^{2}}{4\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{abc}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^2/(d*x^2+c)^3,x)
[Out]
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Maxima [A] time = 1.33405, size = 117, normalized size = 1.75 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} + \frac{b^{2} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233684, size = 146, normalized size = 2.18 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.96983, size = 87, normalized size = 1.3 \[ \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x^{2} \left (4 a b d^{2} - 4 b^{2} c d\right )}{4 c^{2} d^{3} + 8 c d^{4} x^{2} + 4 d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.231473, size = 103, normalized size = 1.54 \[ \frac{b^{2}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x^{2} + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{4 \,{\left (d x^{2} + c\right )}^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x/(d*x^2 + c)^3,x, algorithm="giac")
[Out]