Optimal. Leaf size=99 \[ \frac{b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b^2 (A b-a B)}{a^4 x}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]
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Rubi [A] time = 0.167147, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b^2 (A b-a B)}{a^4 x}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^8*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 23.9656, size = 85, normalized size = 0.86 \[ - \frac{A}{7 a x^{7}} + \frac{A b - B a}{5 a^{2} x^{5}} - \frac{b \left (A b - B a\right )}{3 a^{3} x^{3}} + \frac{b^{2} \left (A b - B a\right )}{a^{4} x} + \frac{b^{\frac{5}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**8/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.114859, size = 101, normalized size = 1.02 \[ -\frac{b^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}-\frac{b^2 (a B-A b)}{a^4 x}+\frac{b (a B-A b)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^8*(a + b*x^2)),x]
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Maple [A] time = 0.01, size = 120, normalized size = 1.2 \[ -{\frac{A}{7\,a{x}^{7}}}+{\frac{Ab}{5\,{a}^{2}{x}^{5}}}-{\frac{B}{5\,a{x}^{5}}}-{\frac{{b}^{2}A}{3\,{a}^{3}{x}^{3}}}+{\frac{Bb}{3\,{a}^{2}{x}^{3}}}+{\frac{{b}^{3}A}{{a}^{4}x}}-{\frac{B{b}^{2}}{{a}^{3}x}}+{\frac{A{b}^{4}}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}B}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^8/(b*x^2+a),x)
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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)*x^8),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247391, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 210 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} - 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{4} + 30 \, A a^{3} + 42 \,{\left (B a^{3} - A a^{2} b\right )} x^{2}}{210 \, a^{4} x^{7}}, -\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \,{\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.66796, size = 187, normalized size = 1.89 \[ \frac{\sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac{15 A a^{3} + x^{6} \left (- 105 A b^{3} + 105 B a b^{2}\right ) + x^{4} \left (35 A a b^{2} - 35 B a^{2} b\right ) + x^{2} \left (- 21 A a^{2} b + 21 B a^{3}\right )}{105 a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**8/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.222384, size = 143, normalized size = 1.44 \[ -\frac{{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{105 \, B a b^{2} x^{6} - 105 \, A b^{3} x^{6} - 35 \, B a^{2} b x^{4} + 35 \, A a b^{2} x^{4} + 21 \, B a^{3} x^{2} - 21 \, A a^{2} b x^{2} + 15 \, A a^{3}}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^8),x, algorithm="giac")
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