Optimal. Leaf size=113 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]
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Rubi [A] time = 0.341636, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}-\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 74.3224, size = 102, normalized size = 0.9 \[ - \frac{A}{5 a^{2} x^{5}} + \frac{2 A b - B a}{3 a^{3} x^{3}} - \frac{b^{2} x \left (A b - B a\right )}{2 a^{4} \left (a + b x^{2}\right )} - \frac{b \left (3 A b - 2 B a\right )}{a^{4} x} - \frac{b^{\frac{3}{2}} \left (7 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**6/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.133975, size = 112, normalized size = 0.99 \[ \frac{b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}+\frac{b^2 x (a B-A b)}{2 a^4 \left (a+b x^2\right )}+\frac{b (2 a B-3 A b)}{a^4 x}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{5 a^2 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 136, normalized size = 1.2 \[ -{\frac{A}{5\,{a}^{2}{x}^{5}}}+{\frac{2\,Ab}{3\,{a}^{3}{x}^{3}}}-{\frac{B}{3\,{a}^{2}{x}^{3}}}-3\,{\frac{{b}^{2}A}{{a}^{4}x}}+2\,{\frac{Bb}{{a}^{3}x}}-{\frac{{b}^{3}xA}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}Bx}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,A{b}^{3}}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,B{b}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^6/(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((B*x^2 + A)/((b*x^2 + a)^2*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240682, size = 1, normalized size = 0.01 \[ \left [\frac{30 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 12 \, A a^{3} - 4 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} - 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{60 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, \frac{15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{30 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.05275, size = 218, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{- 6 A a^{3} + x^{6} \left (- 105 A b^{3} + 75 B a b^{2}\right ) + x^{4} \left (- 70 A a b^{2} + 50 B a^{2} b\right ) + x^{2} \left (14 A a^{2} b - 10 B a^{3}\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**6/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.229561, size = 151, normalized size = 1.34 \[ \frac{{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{B a b^{2} x - A b^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{4}} + \frac{30 \, B a b x^{4} - 45 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^6),x, algorithm="giac")
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