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]

-1/3*b/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 = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.78, size = 54, normalized size = 1.23 \[ -\frac {{\left (3 \, a x^{6} + 2 \, b x^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, {\left (a^{4} x^{8} + 2 \, a^{3} b x^{4} + a^{2} b^{2}\right )}} \]

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]

-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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giac [A]  time = 0.19, size = 33, normalized size = 0.75 \[ -\frac {3 \, a x^{4} + 2 \, b}{6 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}} a^{2}} + \frac {1}{3 \, a^{2} \sqrt {b}} \]

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]

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

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maple [A]  time = 0.01, size = 39, normalized size = 0.89 \[ -\frac {\left (a \,x^{4}+b \right ) \left (3 a \,x^{4}+2 b \right )}{6 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} a^{2} x^{10}} \]

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 = 0.87, size = 33, normalized size = 0.75 \[ -\frac {3 \, {\left (a + \frac {b}{x^{4}}\right )} x^{4} - b}{6 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{2} x^{6}} \]

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]

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

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mupad [B]  time = 1.27, size = 36, normalized size = 0.82 \[ -\frac {x^2\,\sqrt {a+\frac {b}{x^4}}\,\left (3\,a\,x^4+2\,b\right )}{6\,a^2\,{\left (a\,x^4+b\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^4)^(5/2)),x)

[Out]

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

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sympy [B]  time = 1.92, 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]

-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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