Optimal. Leaf size=42 \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]
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Rubi [A] time = 0.0384338, antiderivative size = 42, 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 \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 4.06167, size = 37, normalized size = 0.88 \[ - \frac{1}{2 a x^{2} \sqrt{a + b x^{4}}} - \frac{b x^{2}}{a^{2} \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0245545, size = 29, normalized size = 0.69 \[ -\frac{a+2 b x^4}{2 a^2 x^2 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 26, normalized size = 0.6 \[ -{\frac{2\,b{x}^{4}+a}{2\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^4+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.44188, size = 49, normalized size = 1.17 \[ -\frac{b x^{2}}{2 \, \sqrt{b x^{4} + a} a^{2}} - \frac{\sqrt{b x^{4} + a}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253297, size = 50, normalized size = 1.19 \[ -\frac{{\left (2 \, b x^{4} + a\right )} \sqrt{b x^{4} + a}}{2 \,{\left (a^{2} b x^{6} + a^{3} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.85669, size = 46, normalized size = 1.1 \[ - \frac{1}{2 a \sqrt{b} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}} - \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{4}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.24261, size = 47, normalized size = 1.12 \[ -\frac{\sqrt{b + \frac{a}{x^{4}}}}{2 \, a^{2}} + \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^3),x, algorithm="giac")
[Out]