Optimal. Leaf size=64 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}+\frac{c (b c-2 a d)}{a^2 x}-\frac{c^2}{3 a x^3} \]
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Rubi [A] time = 0.133459, antiderivative size = 64, 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 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}+\frac{c (b c-2 a d)}{a^2 x}-\frac{c^2}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^2/(x^4*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 21.0205, size = 56, normalized size = 0.88 \[ - \frac{c^{2}}{3 a x^{3}} - \frac{c \left (2 a d - b c\right )}{a^{2} x} + \frac{\left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**2/x**4/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.112273, size = 66, normalized size = 1.03 \[ \frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c (2 a d-b c)}{a^2 x}-\frac{c^2}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^2/(x^4*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 98, normalized size = 1.5 \[ -{\frac{{c}^{2}}{3\,a{x}^{3}}}-2\,{\frac{cd}{ax}}+{\frac{b{c}^{2}}{{a}^{2}x}}+{{d}^{2}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{bcd}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{2}}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^2/x^4/(b*x^2+a),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((d*x^2 + c)^2/((b*x^2 + a)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241347, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (a c^{2} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a^{2} x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (a c^{2} - 3 \,{\left (b c^{2} - 2 \, a c d\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.10921, size = 172, normalized size = 2.69 \[ - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac{a c^{2} + x^{2} \left (6 a c d - 3 b c^{2}\right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**2/x**4/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.224712, size = 97, normalized size = 1.52 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)*x^4),x, algorithm="giac")
[Out]