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

Optimal. Leaf size=44 \[ \frac{b \sqrt{a+b x^4}}{3 a^2 x^2}-\frac{\sqrt{a+b x^4}}{6 a x^6} \]

[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 = 0.0101492, 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} \[ \frac{b \sqrt{a+b x^4}}{3 a^2 x^2}-\frac{\sqrt{a+b x^4}}{6 a x^6} \]

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 = 0.0066676, size = 29, normalized size = 0.66 \[ -\frac{\left (a-2 b x^4\right ) \sqrt{a+b x^4}}{6 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[a + b*x^4]),x]

[Out]

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

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Maple [A]  time = 0.006, size = 26, 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 = 0.953857, size = 47, 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 = 1.44623, size = 61, normalized size = 1.39 \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 = 1.09456, size = 44, normalized size = 1. \begin{align*} - \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}} \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]

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

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Giac [A]  time = 1.58475, size = 36, normalized size = 0.82 \begin{align*} -\frac{{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{4}}} b}{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*((b + a/x^4)^(3/2) - 3*sqrt(b + a/x^4)*b)/a^2