Optimal. Leaf size=44 \[ -\frac{b}{3 a^2 x^6 \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{1}{2 a x^2 \left (a+\frac{b}{x^4}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0656855, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{b}{3 a^2 x^6 \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{1}{2 a x^2 \left (a+\frac{b}{x^4}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^4)^(5/2)*x^3),x]
[Out]
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Rubi in Sympy [A] time = 4.39542, size = 39, normalized size = 0.89 \[ - \frac{1}{2 a x^{2} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{b}{3 a^{2} x^{6} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**4)**(5/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0299482, size = 40, normalized size = 0.91 \[ \frac{-3 a x^4-2 b}{6 a^2 x^2 \sqrt{a+\frac{b}{x^4}} \left (a x^4+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^4)^(5/2)*x^3),x]
[Out]
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Maple [A] time = 0.01, size = 39, normalized size = 0.9 \[ -{\frac{ \left ( a{x}^{4}+b \right ) \left ( 3\,a{x}^{4}+2\,b \right ) }{6\,{a}^{2}{x}^{10}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^4)^(5/2)/x^3,x)
[Out]
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Maxima [A] time = 1.46495, size = 45, normalized size = 1.02 \[ -\frac{3 \,{\left (a + \frac{b}{x^{4}}\right )} x^{4} - b}{6 \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^4)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24514, size = 73, normalized size = 1.66 \[ -\frac{{\left (3 \, a x^{6} + 2 \, b x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \,{\left (a^{4} x^{8} + 2 \, a^{3} b x^{4} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^4)^(5/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.6066, size = 105, normalized size = 2.39 \[ - \frac{3 a x^{4}}{6 a^{3} \sqrt{b} x^{4} \sqrt{\frac{a x^{4}}{b} + 1} + 6 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{2 b}{6 a^{3} \sqrt{b} x^{4} \sqrt{\frac{a x^{4}}{b} + 1} + 6 a^{2} b^{\frac{3}{2}} \sqrt{\frac{a x^{4}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**4)**(5/2)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^4)^(5/2)*x^3),x, algorithm="giac")
[Out]