Optimal. Leaf size=142 \[ -\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{b^2 x (15 A b-11 a B)}{8 a^5 \left (a+b x^2\right )}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 x (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{A}{5 a^3 x^5} \]
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Rubi [A] time = 0.590916, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{b^2 x (15 A b-11 a B)}{8 a^5 \left (a+b x^2\right )}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 x (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{A}{5 a^3 x^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**6/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.169057, size = 139, normalized size = 0.98 \[ \frac{7 b^{3/2} (5 a B-9 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{-8 a^4 \left (3 A+5 B x^2\right )+8 a^3 b x^2 \left (9 A+35 B x^2\right )+7 a^2 b^2 x^4 \left (125 B x^2-72 A\right )+525 a b^3 x^6 \left (B x^2-3 A\right )-945 A b^4 x^8}{120 a^5 x^5 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.02, size = 177, normalized size = 1.3 \[ -{\frac{A}{5\,{a}^{3}{x}^{5}}}+{\frac{Ab}{{x}^{3}{a}^{4}}}-{\frac{B}{3\,{x}^{3}{a}^{3}}}-6\,{\frac{{b}^{2}A}{{a}^{5}x}}+3\,{\frac{Bb}{{a}^{4}x}}-{\frac{15\,A{b}^{4}{x}^{3}}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,B{b}^{3}{x}^{3}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}Ax}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}Bx}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{b}^{3}A}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,B{b}^{2}}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^6/(b*x^2+a)^3,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)^3*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.250722, size = 1, normalized size = 0.01 \[ \left [\frac{210 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 350 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} - 48 \, A a^{4} + 112 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 16 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} - 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{240 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, \frac{105 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 175 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} - 24 \, A a^{4} + 56 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 8 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} + 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{120 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.11975, size = 260, normalized size = 1.83 \[ - \frac{7 \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right ) \log{\left (- \frac{7 a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right )}{- 63 A b^{3} + 35 B a b^{2}} + x \right )}}{16} + \frac{7 \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right ) \log{\left (\frac{7 a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right )}{- 63 A b^{3} + 35 B a b^{2}} + x \right )}}{16} + \frac{- 24 A a^{4} + x^{8} \left (- 945 A b^{4} + 525 B a b^{3}\right ) + x^{6} \left (- 1575 A a b^{3} + 875 B a^{2} b^{2}\right ) + x^{4} \left (- 504 A a^{2} b^{2} + 280 B a^{3} b\right ) + x^{2} \left (72 A a^{3} b - 40 B a^{4}\right )}{120 a^{7} x^{5} + 240 a^{6} b x^{7} + 120 a^{5} b^{2} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**6/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.263786, size = 182, normalized size = 1.28 \[ \frac{7 \,{\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} + \frac{11 \, B a b^{3} x^{3} - 15 \, A b^{4} x^{3} + 13 \, B a^{2} b^{2} x - 17 \, A a b^{3} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{5}} + \frac{45 \, B a b x^{4} - 90 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 15 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{5} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*x^6),x, algorithm="giac")
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