Optimal. Leaf size=55 \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]
[Out]
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Rubi [A] time = 0.115327, antiderivative size = 55, 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 \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^2*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{c x} + \frac{\int b^{2}\, dx}{d} - \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0739474, size = 55, normalized size = 1. \[ -\frac{a^2}{c x}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} d^{3/2}}+\frac{b^2 x}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.007, size = 85, normalized size = 1.6 \[{\frac{{b}^{2}x}{d}}-{\frac{{a}^{2}d}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab}{\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{2}c}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{cx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^2/(d*x^2+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231899, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \log \left (-\frac{2 \, c d x -{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (b^{2} c x^{2} - a^{2} d\right )} \sqrt{-c d}}{2 \, \sqrt{-c d} c d x}, -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (b^{2} c x^{2} - a^{2} d\right )} \sqrt{c d}}{\sqrt{c d} c d x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.88532, size = 165, normalized size = 3. \[ - \frac{a^{2}}{c x} + \frac{b^{2} x}{d} + \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.234251, size = 85, normalized size = 1.55 \[ \frac{b^{2} x}{d} - \frac{a^{2}}{c x} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^2),x, algorithm="giac")
[Out]