Optimal. Leaf size=57 \[ -\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^3}+\frac{2 a \left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.088691, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^2 \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b^3}+\frac{2 a \left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^2]*x^7),x]
[Out]
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Rubi in Sympy [A] time = 10.5054, size = 49, normalized size = 0.86 \[ - \frac{a^{2} \sqrt{a + \frac{b}{x^{2}}}}{b^{3}} + \frac{2 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3 b^{3}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{5 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(1/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.0444389, size = 42, normalized size = 0.74 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (8 a^2 x^4-4 a b x^2+3 b^2\right )}{15 b^3 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^2]*x^7),x]
[Out]
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Maple [A] time = 0.01, size = 50, normalized size = 0.9 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{x}^{4}{a}^{2}-4\,ab{x}^{2}+3\,{b}^{2} \right ) }{15\,{b}^{3}{x}^{6}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(1/2)/x^7,x)
[Out]
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Maxima [A] time = 1.43598, size = 63, normalized size = 1.11 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b^{3}} + \frac{2 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} a}{3 \, b^{3}} - \frac{\sqrt{a + \frac{b}{x^{2}}} a^{2}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238032, size = 57, normalized size = 1. \[ -\frac{{\left (8 \, a^{2} x^{4} - 4 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{15 \, b^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.62606, size = 750, normalized size = 13.16 \[ - \frac{8 a^{\frac{15}{2}} b^{\frac{9}{2}} x^{10} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{20 a^{\frac{13}{2}} b^{\frac{11}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{15 a^{\frac{11}{2}} b^{\frac{13}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{5 a^{\frac{9}{2}} b^{\frac{15}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{5 a^{\frac{7}{2}} b^{\frac{17}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} - \frac{3 a^{\frac{5}{2}} b^{\frac{19}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{8 a^{8} b^{4} x^{11}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{24 a^{7} b^{5} x^{9}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{24 a^{6} b^{6} x^{7}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} + \frac{8 a^{5} b^{7} x^{5}}{15 a^{\frac{11}{2}} b^{7} x^{11} + 45 a^{\frac{9}{2}} b^{8} x^{9} + 45 a^{\frac{7}{2}} b^{9} x^{7} + 15 a^{\frac{5}{2}} b^{10} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(1/2)/x**7,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^7),x, algorithm="giac")
[Out]