Optimal. Leaf size=88 \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]
[Out]
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Rubi [A] time = 0.100294, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b/x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.9117, size = 82, normalized size = 0.93 \[ \frac{x^{5}}{5 a \sqrt{a + \frac{b}{x^{2}}}} - \frac{2 b x^{3}}{5 a^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{8 b^{2} x}{5 a^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{16 b^{2} x \sqrt{a + \frac{b}{x^{2}}}}{5 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(a+b/x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0338398, size = 52, normalized size = 0.59 \[ \frac{a^3 x^6-2 a^2 b x^4+8 a b^2 x^2+16 b^3}{5 a^4 x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b/x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 60, normalized size = 0.7 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{3}{x}^{6}-2\,{a}^{2}b{x}^{4}+8\,a{b}^{2}{x}^{2}+16\,{b}^{3} \right ) }{5\,{a}^{4}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(a+b/x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.44566, size = 93, normalized size = 1.06 \[ \frac{b^{3}}{\sqrt{a + \frac{b}{x^{2}}} a^{4} x} + \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 5 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b x^{3} + 15 \, \sqrt{a + \frac{b}{x^{2}}} b^{2} x}{5 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239347, size = 84, normalized size = 0.95 \[ \frac{{\left (a^{3} x^{7} - 2 \, a^{2} b x^{5} + 8 \, a b^{2} x^{3} + 16 \, b^{3} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.09096, size = 337, normalized size = 3.83 \[ \frac{a^{5} b^{\frac{19}{2}} x^{10} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{5 a^{3} b^{\frac{23}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{30 a^{2} b^{\frac{25}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{40 a b^{\frac{27}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{16 b^{\frac{29}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(a+b/x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="giac")
[Out]