Optimal. Leaf size=58 \[ -\frac{a^2}{2 c x^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}+\frac{a \log (x) (2 b c-a d)}{c^2} \]
[Out]
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Rubi [A] time = 0.149173, antiderivative size = 58, 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 \[ -\frac{a^2}{2 c x^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}+\frac{a \log (x) (2 b c-a d)}{c^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^3*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 22.9969, size = 53, normalized size = 0.91 \[ - \frac{a^{2}}{2 c x^{2}} - \frac{a \left (a d - 2 b c\right ) \log{\left (x^{2} \right )}}{2 c^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**3/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0473834, size = 60, normalized size = 1.03 \[ \frac{a^2 (-c) d-2 a d x^2 \log (x) (a d-2 b c)+x^2 (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 81, normalized size = 1.4 \[ -{\frac{{a}^{2}}{2\,c{x}^{2}}}-{\frac{\ln \left ( x \right ){a}^{2}d}{{c}^{2}}}+2\,{\frac{a\ln \left ( x \right ) b}{c}}+{\frac{d\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{2}}}-{\frac{\ln \left ( d{x}^{2}+c \right ) ab}{c}}+{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^3/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.36137, size = 95, normalized size = 1.64 \[ \frac{{\left (2 \, a b c - a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{2}} - \frac{a^{2}}{2 \, c x^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239451, size = 100, normalized size = 1.72 \[ -\frac{a^{2} c d -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (d x^{2} + c\right ) - 2 \,{\left (2 \, a b c d - a^{2} d^{2}\right )} x^{2} \log \left (x\right )}{2 \, c^{2} d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.63598, size = 49, normalized size = 0.84 \[ - \frac{a^{2}}{2 c x^{2}} - \frac{a \left (a d - 2 b c\right ) \log{\left (x \right )}}{c^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**3/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.228465, size = 123, normalized size = 2.12 \[ \frac{{\left (2 \, a b c - a^{2} d\right )}{\rm ln}\left (x^{2}\right )}{2 \, c^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d} - \frac{2 \, a b c x^{2} - a^{2} d x^{2} + a^{2} c}{2 \, c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^3),x, algorithm="giac")
[Out]