Optimal. Leaf size=66 \[ \frac{8 b^2 x \sqrt{a+\frac{b}{x^2}}}{15 a^3}-\frac{4 b x^3 \sqrt{a+\frac{b}{x^2}}}{15 a^2}+\frac{x^5 \sqrt{a+\frac{b}{x^2}}}{5 a} \]
[Out]
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Rubi [A] time = 0.0777975, antiderivative size = 66, 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 \[ \frac{8 b^2 x \sqrt{a+\frac{b}{x^2}}}{15 a^3}-\frac{4 b x^3 \sqrt{a+\frac{b}{x^2}}}{15 a^2}+\frac{x^5 \sqrt{a+\frac{b}{x^2}}}{5 a} \]
Antiderivative was successfully verified.
[In] Int[x^4/Sqrt[a + b/x^2],x]
[Out]
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Rubi in Sympy [A] time = 6.09426, size = 60, normalized size = 0.91 \[ \frac{x^{5} \sqrt{a + \frac{b}{x^{2}}}}{5 a} - \frac{4 b x^{3} \sqrt{a + \frac{b}{x^{2}}}}{15 a^{2}} + \frac{8 b^{2} x \sqrt{a + \frac{b}{x^{2}}}}{15 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0378687, size = 40, normalized size = 0.61 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (3 a^2 x^4-4 a b x^2+8 b^2\right )}{15 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/Sqrt[a + b/x^2],x]
[Out]
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Maple [A] time = 0.008, size = 50, normalized size = 0.8 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 3\,{x}^{4}{a}^{2}-4\,ab{x}^{2}+8\,{b}^{2} \right ) }{15\,{a}^{3}x}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(a+b/x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.45435, size = 68, normalized size = 1.03 \[ \frac{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 10 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b x^{3} + 15 \, \sqrt{a + \frac{b}{x^{2}}} b^{2} x}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(a + b/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232059, size = 54, normalized size = 0.82 \[ \frac{{\left (3 \, a^{2} x^{5} - 4 \, a b x^{3} + 8 \, b^{2} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(a + b/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.02473, size = 279, normalized size = 4.23 \[ \frac{3 a^{4} b^{\frac{9}{2}} x^{8} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{2 a^{3} b^{\frac{11}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{3 a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{12 a b^{\frac{15}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} + \frac{8 b^{\frac{17}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{2} + 15 a^{3} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a + \frac{b}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/sqrt(a + b/x^2),x, algorithm="giac")
[Out]