Optimal. Leaf size=75 \[ -\frac{a^2}{4 c x^4}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^3}+\frac{\log (x) (b c-a d)^2}{c^3}-\frac{a (2 b c-a d)}{2 c^2 x^2} \]
[Out]
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Rubi [A] time = 0.165956, antiderivative size = 75, 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 \[ -\frac{a^2}{4 c x^4}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^3}+\frac{\log (x) (b c-a d)^2}{c^3}-\frac{a (2 b c-a d)}{2 c^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 26.7758, size = 68, normalized size = 0.91 \[ - \frac{a^{2}}{4 c x^{4}} + \frac{a \left (a d - 2 b c\right )}{2 c^{2} x^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (x^{2} \right )}}{2 c^{3}} - \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**5/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0752354, size = 72, normalized size = 0.96 \[ -\frac{-4 x^4 \log (x) (b c-a d)^2+a c \left (a c-2 a d x^2+4 b c x^2\right )+2 x^4 (b c-a d)^2 \log \left (c+d x^2\right )}{4 c^3 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.011, size = 116, normalized size = 1.6 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}}}+{\frac{\ln \left ( x \right ){a}^{2}{d}^{2}}{{c}^{3}}}-2\,{\frac{\ln \left ( x \right ) abd}{{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{c}}+{\frac{{a}^{2}d}{2\,{c}^{2}{x}^{2}}}-{\frac{ab}{c{x}^{2}}}-{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}{d}^{2}}{2\,{c}^{3}}}+{\frac{\ln \left ( d{x}^{2}+c \right ) abd}{{c}^{2}}}-{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^5/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.32746, size = 130, normalized size = 1.73 \[ -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac{a^{2} c + 2 \,{\left (2 \, a b c - a^{2} d\right )} x^{2}}{4 \, c^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229185, size = 132, normalized size = 1.76 \[ -\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (d x^{2} + c\right ) - 4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} \log \left (x\right ) + a^{2} c^{2} + 2 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{4 \, c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.30967, size = 66, normalized size = 0.88 \[ \frac{- a^{2} c + x^{2} \left (2 a^{2} d - 4 a b c\right )}{4 c^{2} x^{4}} + \frac{\left (a d - b c\right )^{2} \log{\left (x \right )}}{c^{3}} - \frac{\left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**5/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.225543, size = 188, normalized size = 2.51 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, c^{3}} - \frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3} d} - \frac{3 \, b^{2} c^{2} x^{4} - 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} + 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^5),x, algorithm="giac")
[Out]