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

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 = 0.0102541, 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, Rules used = {271, 264} \[ -\frac{b x^2}{a^2 \sqrt{a+b x^4}}-\frac{1}{2 a x^2 \sqrt{a+b x^4}} \]

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])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^4\right )^{3/2}} \, dx &=-\frac{1}{2 a x^2 \sqrt{a+b x^4}}-\frac{(2 b) \int \frac{x}{\left (a+b x^4\right )^{3/2}} \, dx}{a}\\ &=-\frac{1}{2 a x^2 \sqrt{a+b x^4}}-\frac{b x^2}{a^2 \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0069317, 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 = 0.004, size = 26, normalized size = 0.6 \begin{align*} -{\frac{2\,b{x}^{4}+a}{2\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

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 = 0.958572, size = 49, normalized size = 1.17 \begin{align*} -\frac{b x^{2}}{2 \, \sqrt{b x^{4} + a} a^{2}} - \frac{\sqrt{b x^{4} + a}}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^4+a)^(3/2),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 = 1.48744, size = 78, normalized size = 1.86 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^4+a)^(3/2),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 = 1.01501, size = 46, normalized size = 1.1 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.16058, size = 47, normalized size = 1.12 \begin{align*} -\frac{\sqrt{b + \frac{a}{x^{4}}}}{2 \, a^{2}} + \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^4+a)^(3/2),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)