Optimal. Leaf size=51 \[ \frac{a^2 \log (x)}{c}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac{b^2 x^2}{2 d} \]
[Out]
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Rubi [A] time = 0.12011, antiderivative size = 51, 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 \[ \frac{a^2 \log (x)}{c}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac{b^2 x^2}{2 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \log{\left (x^{2} \right )}}{2 c} + \frac{\int ^{x^{2}} b^{2}\, dx}{2 d} - \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.036119, size = 50, normalized size = 0.98 \[ \frac{2 a^2 d^2 \log (x)-(b c-a d)^2 \log \left (c+d x^2\right )+b^2 c d x^2}{2 c d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.008, size = 69, normalized size = 1.4 \[{\frac{{b}^{2}{x}^{2}}{2\,d}}+{\frac{{a}^{2}\ln \left ( x \right ) }{c}}-{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,c}}+{\frac{\ln \left ( d{x}^{2}+c \right ) ab}{d}}-{\frac{c\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.3582, size = 82, normalized size = 1.61 \[ \frac{b^{2} x^{2}}{2 \, d} + \frac{a^{2} \log \left (x^{2}\right )}{2 \, c} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235367, size = 80, normalized size = 1.57 \[ \frac{b^{2} c d x^{2} + 2 \, a^{2} d^{2} \log \left (x\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.13265, size = 41, normalized size = 0.8 \[ \frac{a^{2} \log{\left (x \right )}}{c} + \frac{b^{2} x^{2}}{2 d} - \frac{\left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.229867, size = 84, normalized size = 1.65 \[ \frac{b^{2} x^{2}}{2 \, d} + \frac{a^{2}{\rm ln}\left (x^{2}\right )}{2 \, c} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x),x, algorithm="giac")
[Out]