3.839 \(\int \frac{1}{x^7 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{b \sqrt{a-b x^4}}{3 a^2 x^2}-\frac{\sqrt{a-b x^4}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.0109992, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {271, 264} \[ -\frac{b \sqrt{a-b x^4}}{3 a^2 x^2}-\frac{\sqrt{a-b x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(6*a*x^6) - (b*Sqrt[a - b*x^4])/(3*a^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}{x^7 \sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{6 a x^6}+\frac{(2 b) \int \frac{1}{x^3 \sqrt{a-b x^4}} \, dx}{3 a}\\ &=-\frac{\sqrt{a-b x^4}}{6 a x^6}-\frac{b \sqrt{a-b x^4}}{3 a^2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0073376, size = 30, normalized size = 0.65 \[ -\frac{\sqrt{a-b x^4} \left (a+2 b x^4\right )}{6 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[a - b*x^4]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \begin{align*} -{\frac{2\,b{x}^{4}+a}{6\,{a}^{2}{x}^{6}}\sqrt{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(-b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.965735, size = 49, normalized size = 1.07 \begin{align*} -\frac{\frac{3 \, \sqrt{-b x^{4} + a} b}{x^{2}} + \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49493, size = 63, normalized size = 1.37 \begin{align*} -\frac{{\left (2 \, b x^{4} + a\right )} \sqrt{-b x^{4} + a}}{6 \, a^{2} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.17731, size = 192, normalized size = 4.17 \begin{align*} \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{6 a x^{4}} - \frac{b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{4}} - 1}}{3 a^{2}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x^{4}}\right |} > 1 \\\frac{i a^{2} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} + \frac{i a b^{\frac{5}{2}} x^{4} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} - \frac{2 i b^{\frac{7}{2}} x^{8} \sqrt{- \frac{a}{b x^{4}} + 1}}{- 6 a^{3} b x^{4} + 6 a^{2} b^{2} x^{8}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(6*a*x**4) - b**(3/2)*sqrt(a/(b*x**4) - 1)/(3*a**2), Abs(a)/(Abs(b)*A
bs(x**4)) > 1), (I*a**2*b**(3/2)*sqrt(-a/(b*x**4) + 1)/(-6*a**3*b*x**4 + 6*a**2*b**2*x**8) + I*a*b**(5/2)*x**4
*sqrt(-a/(b*x**4) + 1)/(-6*a**3*b*x**4 + 6*a**2*b**2*x**8) - 2*I*b**(7/2)*x**8*sqrt(-a/(b*x**4) + 1)/(-6*a**3*
b*x**4 + 6*a**2*b**2*x**8), True))

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Giac [A]  time = 1.09041, size = 42, normalized size = 0.91 \begin{align*} -\frac{3 \, b \sqrt{-b + \frac{a}{x^{4}}} +{\left (-b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}}}{6 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

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