Optimal. Leaf size=127 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{x \left (3 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{6 a^3 \left (a+b x^2\right )}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.363003, antiderivative size = 125, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{x \left (\frac{b c (5 b c-6 a d)}{a^2}+3 d^2\right )}{6 a \left (a+b x^2\right )}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^2/(x^4*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 42.1185, size = 110, normalized size = 0.87 \[ - \frac{c^{2}}{3 a x^{3} \left (a + b x^{2}\right )} - \frac{c \left (6 a d - 5 b c\right )}{3 a^{3} x} + \frac{x \left (3 a^{2} d^{2} - b c \left (6 a d - 5 b c\right )\right )}{6 a^{3} \left (a + b x^{2}\right )} + \frac{\left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**2/x**4/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.113971, size = 107, normalized size = 0.84 \[ \frac{x (a d-b c)^2}{2 a^3 \left (a+b x^2\right )}-\frac{2 c (a d-b c)}{a^3 x}-\frac{c^2}{3 a^2 x^3}+\frac{\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^2/(x^4*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 161, normalized size = 1.3 \[ -{\frac{{c}^{2}}{3\,{a}^{2}{x}^{3}}}-2\,{\frac{cd}{{a}^{2}x}}+2\,{\frac{b{c}^{2}}{{a}^{3}x}}+{\frac{x{d}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{cxbd}{{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}{c}^{2}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{bdc}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{5\,{b}^{2}{c}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^2/x^4/(b*x^2+a)^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((d*x^2 + c)^2/((b*x^2 + a)^2*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.245404, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 6 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b}}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \,{\left (5 \, a b c^{2} - 6 \, a^{2} c d\right )} x^{2}\right )} \sqrt{a b}}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.74846, size = 248, normalized size = 1.95 \[ - \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (3 a^{2} d^{2} - 18 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 12 a^{2} c d + 10 a b c^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**2/x**4/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.248913, size = 151, normalized size = 1.19 \[ \frac{{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2/((b*x^2 + a)^2*x^4),x, algorithm="giac")
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