3.857 \(\int \frac{1}{x^3 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]

[Out]

-1/(2*a*x^2*Sqrt[a + b*x^4]) - (b*x^2)/(a^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]

-1/(2*a*x^2*Sqrt[a + b*x^4]) - (b*x^2)/(a^2*Sqrt[a + b*x^4])

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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]

-1/(2*a*x**2*sqrt(a + b*x**4)) - b*x**2/(a**2*sqrt(a + b*x**4))

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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]

-(a + 2*b*x^4)/(2*a^2*x^2*Sqrt[a + b*x^4])

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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]

-1/2*(2*b*x^4+a)/x^2/(b*x^4+a)^(1/2)/a^2

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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]

-1/2*b*x^2/(sqrt(b*x^4 + a)*a^2) - 1/2*sqrt(b*x^4 + a)/(a^2*x^2)

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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]

-1/2*(2*b*x^4 + a)*sqrt(b*x^4 + a)/(a^2*b*x^6 + a^3*x^2)

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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]

-1/(2*a*sqrt(b)*x**4*sqrt(a/(b*x**4) + 1)) - sqrt(b)/(a**2*sqrt(a/(b*x**4) + 1))

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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]

-1/2*sqrt(b + a/x^4)/a^2 + 1/256*x^2/(sqrt(b*x^4 + a)*a^3*b^3)