Optimal. Leaf size=66 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}-\frac{a (2 b c-a d)}{c^2 x} \]
[Out]
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Rubi [A] time = 0.12916, antiderivative size = 66, 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 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}-\frac{a (2 b c-a d)}{c^2 x} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^4*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 21.3941, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{3 c x^{3}} + \frac{a \left (a d - 2 b c\right )}{c^{2} x} + \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{5}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**4/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0969104, size = 64, normalized size = 0.97 \[ -\frac{a^2}{3 c x^3}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}+\frac{a (a d-2 b c)}{c^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 98, normalized size = 1.5 \[ -{\frac{{a}^{2}}{3\,c{x}^{3}}}+{\frac{{a}^{2}d}{{c}^{2}x}}-2\,{\frac{ab}{cx}}+{\frac{{a}^{2}{d}^{2}}{{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{abd}{c\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{{b}^{2}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^4/(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^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235958, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) - 2 \,{\left (a^{2} c + 3 \,{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{-c d}}{6 \, \sqrt{-c d} c^{2} x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (a^{2} c + 3 \,{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{c d}}{3 \, \sqrt{c d} c^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.13898, size = 172, normalized size = 2.61 \[ - \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{- a^{2} c + x^{2} \left (3 a^{2} d - 6 a b c\right )}{3 c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**4/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.223582, size = 96, normalized size = 1.45 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c^{2}} - \frac{6 \, a b c x^{2} - 3 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^4),x, algorithm="giac")
[Out]