Optimal. Leaf size=87 \[ -\frac{a^2}{5 c x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}}-\frac{(b c-a d)^2}{c^3 x}-\frac{a (2 b c-a d)}{3 c^2 x^3} \]
[Out]
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Rubi [A] time = 0.151743, antiderivative size = 87, 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 \[ -\frac{a^2}{5 c x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}}-\frac{(b c-a d)^2}{c^3 x}-\frac{a (2 b c-a d)}{3 c^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 27.0113, size = 73, normalized size = 0.84 \[ - \frac{a^{2}}{5 c x^{5}} + \frac{a \left (a d - 2 b c\right )}{3 c^{2} x^{3}} - \frac{\left (a d - b c\right )^{2}}{c^{3} x} - \frac{\sqrt{d} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**6/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.119188, size = 86, normalized size = 0.99 \[ -\frac{a^2}{5 c x^5}-\frac{\sqrt{d} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{7/2}}-\frac{(b c-a d)^2}{c^3 x}+\frac{a (a d-2 b c)}{3 c^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^6*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 143, normalized size = 1.6 \[ -{\frac{{a}^{2}}{5\,c{x}^{5}}}-{\frac{{a}^{2}{d}^{2}}{{c}^{3}x}}+2\,{\frac{abd}{{c}^{2}x}}-{\frac{{b}^{2}}{cx}}+{\frac{{a}^{2}d}{3\,{c}^{2}{x}^{3}}}-{\frac{2\,ab}{3\,c{x}^{3}}}-{\frac{{a}^{2}{d}^{3}}{{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+2\,{\frac{ab{d}^{2}}{{c}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{d{b}^{2}}{c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^6/(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^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242336, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} - 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 30 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 10 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{30 \, c^{3} x^{5}}, -\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \,{\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.91206, size = 207, normalized size = 2.38 \[ \frac{\sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{4} \sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac{\sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{4} \sqrt{- \frac{d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac{3 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 30 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 5 a^{2} c d + 10 a b c^{2}\right )}{15 c^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**6/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.221866, size = 151, normalized size = 1.74 \[ -\frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c^{3}} - \frac{15 \, b^{2} c^{2} x^{4} - 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} + 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, c^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)*x^6),x, algorithm="giac")
[Out]