3.2101 \(\int \frac{1}{(a+\frac{b}{x^4})^{5/2} x^3} \, dx\)

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]

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

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Rubi [A]  time = 0.0143726, 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, Rules used = {271, 264} \[ -\frac{b}{3 a^2 x^6 \left (a+\frac{b}{x^4}\right )^{3/2}}-\frac{1}{2 a x^2 \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^4)^(5/2)*x^3),x]

[Out]

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

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}{\left (a+\frac{b}{x^4}\right )^{5/2} x^3} \, dx &=-\frac{1}{2 a \left (a+\frac{b}{x^4}\right )^{3/2} x^2}+\frac{(2 b) \int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2} x^7} \, dx}{a}\\ &=-\frac{b}{3 a^2 \left (a+\frac{b}{x^4}\right )^{3/2} x^6}-\frac{1}{2 a \left (a+\frac{b}{x^4}\right )^{3/2} x^2}\\ \end{align*}

Mathematica [A]  time = 0.0175852, 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]

(-2*b - 3*a*x^4)/(6*a^2*Sqrt[a + b/x^4]*x^2*(b + a*x^4))

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Maple [A]  time = 0.004, size = 39, normalized size = 0.9 \begin{align*} -{\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^4)^(5/2)/x^3,x)

[Out]

-1/6*(a*x^4+b)*(3*a*x^4+2*b)/a^2/x^10/((a*x^4+b)/x^4)^(5/2)

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Maxima [A]  time = 0.967349, size = 45, normalized size = 1.02 \begin{align*} -\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}} \end{align*}

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]

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

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Fricas [A]  time = 1.49561, size = 111, normalized size = 2.52 \begin{align*} -\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 )}} \end{align*}

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]

-1/6*(3*a*x^6 + 2*b*x^2)*sqrt((a*x^4 + b)/x^4)/(a^4*x^8 + 2*a^3*b*x^4 + a^2*b^2)

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Sympy [B]  time = 2.47464, size = 105, normalized size = 2.39 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**4)**(5/2)/x**3,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}

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]

integrate(1/((a + b/x^4)^(5/2)*x^3), x)