Optimal. Leaf size=126 \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a (6 b c-5 a d)}{3 c^3 x} \]
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Rubi [A] time = 0.336312, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )}+\frac{(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a (6 b c-5 a d)}{3 c^3 x} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^4*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 42.3322, size = 110, normalized size = 0.87 \[ - \frac{a^{2}}{3 c x^{3} \left (c + d x^{2}\right )} + \frac{a \left (5 a d - 6 b c\right )}{3 c^{3} x} + \frac{x \left (a d \left (5 a d - 6 b c\right ) + 3 b^{2} c^{2}\right )}{6 c^{3} \left (c + d x^{2}\right )} + \frac{\left (a d - b c\right ) \left (5 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{7}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**4/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.105999, size = 107, normalized size = 0.85 \[ \frac{\left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{7/2} \sqrt{d}}-\frac{a^2}{3 c^2 x^3}+\frac{x (b c-a d)^2}{2 c^3 \left (c+d x^2\right )}+\frac{2 a (a d-b c)}{c^3 x} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^4*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.016, size = 161, normalized size = 1.3 \[ -{\frac{{a}^{2}}{3\,{c}^{2}{x}^{3}}}+2\,{\frac{{a}^{2}d}{{c}^{3}x}}-2\,{\frac{ab}{{c}^{2}x}}+{\frac{{a}^{2}{d}^{2}x}{2\,{c}^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{xabd}{{c}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{x{b}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}+{\frac{5\,{a}^{2}{d}^{2}}{2\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{abd}{{c}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^4/(d*x^2+c)^2,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)^2*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.228138, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \,{\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-c d}}{12 \,{\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} +{\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (3 \,{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \,{\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{c d}}{6 \,{\left (c^{3} d x^{5} + c^{4} x^{3}\right )} \sqrt{c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.63194, size = 248, normalized size = 1.97 \[ - \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 18 a b c d + 3 b^{2} c^{2}\right ) + x^{2} \left (10 a^{2} c d - 12 a b c^{2}\right )}{6 c^{4} x^{3} + 6 c^{3} d x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**4/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.226281, size = 150, normalized size = 1.19 \[ \frac{{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (d x^{2} + c\right )} c^{3}} - \frac{6 \, a b c x^{2} - 6 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^2*x^4),x, algorithm="giac")
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