Optimal. Leaf size=90 \[ \frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}-\frac{b x^2 (b c-a d)}{d^3}+\frac{b^2 x^4}{4 d^2} \]
[Out]
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Rubi [A] time = 0.257606, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}-\frac{b x^2 (b c-a d)}{d^3}+\frac{b^2 x^4}{4 d^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(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. \[ \frac{b^{2} \int ^{x^{2}} x\, dx}{2 d^{2}} + \frac{b x^{2} \left (a d - b c\right )}{d^{3}} + \frac{c \left (a d - b c\right )^{2}}{2 d^{4} \left (c + d x^{2}\right )} + \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{2 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.10544, size = 87, normalized size = 0.97 \[ \frac{2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )+4 b d x^2 (a d-b c)+\frac{2 c (b c-a d)^2}{c+d x^2}+b^2 d^2 x^4}{4 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x^2)^2)/(c + d*x^2)^2,x]
[Out]
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Maple [A] time = 0.016, size = 142, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{4}}{4\,{d}^{2}}}+{\frac{ab{x}^{2}}{{d}^{2}}}-{\frac{{b}^{2}c{x}^{2}}{{d}^{3}}}+{\frac{{a}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{ab{c}^{2}}{{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}{c}^{3}}{2\,{d}^{4} \left ( d{x}^{2}+c \right ) }}+{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{d}^{2}}}-2\,{\frac{\ln \left ( d{x}^{2}+c \right ) abc}{{d}^{3}}}+{\frac{3\,\ln \left ( d{x}^{2}+c \right ){b}^{2}{c}^{2}}{2\,{d}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x^2+a)^2/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.3477, size = 144, normalized size = 1.6 \[ \frac{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}}{2 \,{\left (d^{5} x^{2} + c d^{4}\right )}} + \frac{b^{2} d x^{4} - 4 \,{\left (b^{2} c - a b d\right )} x^{2}}{4 \, d^{3}} + \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229741, size = 217, normalized size = 2.41 \[ \frac{b^{2} d^{3} x^{6} + 2 \, b^{2} c^{3} - 4 \, a b c^{2} d + 2 \, a^{2} c d^{2} -{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )} x^{4} - 4 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x^{2} + 2 \,{\left (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{2} + c d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.36309, size = 97, normalized size = 1.08 \[ \frac{b^{2} x^{4}}{4 d^{2}} + \frac{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}}{2 c d^{4} + 2 d^{5} x^{2}} + \frac{x^{2} \left (a b d - b^{2} c\right )}{d^{3}} + \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{2 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.224205, size = 220, normalized size = 2.44 \[ \frac{\frac{{\left (d x^{2} + c\right )}^{2}{\left (b^{2} - \frac{2 \,{\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (d x^{2} + c\right )} d}\right )}}{d^{3}} - \frac{2 \,{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left (\frac{{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} + \frac{2 \,{\left (\frac{b^{2} c^{3} d^{2}}{d x^{2} + c} - \frac{2 \, a b c^{2} d^{3}}{d x^{2} + c} + \frac{a^{2} c d^{4}}{d x^{2} + c}\right )}}{d^{5}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*x^3/(d*x^2 + c)^2,x, algorithm="giac")
[Out]